Mathematics: Pre-Algebra: Chapter 6: Basic Algebra

the sum of twice a number and six

2x + 6

One less than twice a number is seventeen.

2x - 1 = 17

Evaluate the expression 3x² + 5xy - 2 for x = 3 and y = -2.

3x² + 5xy - 2 (3)3² + (5)(3)(-2) - 2 (3)(9) + (5)(3)(-2) - 2 27 + 15(-2) - 2 27 + (-30) - 2 -3 - 2 = -5

Plotting Points on a Graph

Cartesian points are written as xy pairs in parentheses, like so: (x, y). To graph a point, first locate its position on the x-axis, then find its location on the y-axis, and finally plot where these meet. The center point of the graph is called the origin and is written as the point (0, 0) because it's located at the zero point on the x-axis and the zero point on the y-axis.

Multiply (2x - 8)(9x + 4)

(2x - 8)(9x + 4) (2x + -8)(9x + 4) (2x)(9x) (2x)(4) (-8)(9x) (-8)(4) 18x² + 8x - 72x - 32 18x² - 72x - 32

Simplify: (2xy + 6x - 2y) - (-3xy + 4x - 7y)

(2xy + 6x - 2y) - (-3xy + 4x - 7y) 5xy + 2x + 5y

Add More than The sum of Added to Increased by

+ addition A number increased by three x + 3

Difference Less than Subtract Reduced by Decreased by

- subtraction Seven less than a number x - 7

This is called a Cartesian coordinate graph. It's made up of two axes ("axes" is just the plural of "axis"):

- The horizontal axis is called the x-axis. - And the vertical one is the y-axis.

Method 2 Add the Negative

-2(7 + -2x) -2(7) + (-2)(-2x) -14 + 4x

Method 1 Leave as Subtraction

-2(7-2x) -2(7) - (-2)(2x) -14 - (-4x) -14 + 4x

Divide -3a²b³ by 12ab.

-3a²b³/12ab = -3aabbb/12ab = -3ab²/12 = -ab²/4

Multiply -3y(x+6y)(3x - 4y)

-3y(x+6y)(3x - 4y) (x)(3x) (x)(-4y) (6y)(3x) (6y)(-4y) 3x² - 4xy + 18xy - 24y² 3x² + 14xy - 24y² (-3y) 3x² + 14xy - 24y² -9x²y - 42xy² + 72y³

Solve for b: -b/-2 = -8

-b/-2 = -8 Be sure you solve for b, not -b. (change to positive and multiply both sides by 2) (b/2)(2) = (-8)(2) b = -16 Don't forget to check your work! -16/-2 = -8 -8 = -8

Solve for x: -x/4 < 3

-x/4 < 3 -x/4(-4) < 3(-4) x < -12 Don't forget to flip that inequality sign when dividing by a negative number. Flip it good. x > -12

We can also add more complicated equations. If we've got this system: 3x + 2y = 7 y = 3x - 2

...then we can add 'em up vertically to get: 3x + 2y + y = 7 + 3x - 2 We'll use this concept to solve systems of linear equations.

Half of a number is twenty.

1/2x = 20

Twelve is the product of a number and three.

12 = 3x

Binomial Expressions

2 Terms Example: xy - 2x

Don't forget to check:

2(-8) - 6 = 5(-8) + 18 -16 - 6 = -40 + 18 -22 = -22

Solve for z: 2/(z + 3) ≠ 2/3z

2/(z + 3) ≠ 2/3z First multiply each side by (z + 3), then multiply each side by (3z). 2/(z + 3)(z + 3) ≠ 2/3z(z + 3) 2 ≠ 2/3z(z + 3) 2(3z) ≠ 2/3z (z + 3) 6z ≠ 2(z + 3) 6z ≠ 2z + 6 6z - 2z ≠ 2z + 6 - 2z 4z ≠ 6 4z/4 ≠ 6/4 z ≠ 3/2 or 1.5

twenty-eight split in half

28 ÷ 2

Method #2

2x - 6 = 12 divide each side by 2 (2x - 6)/2 = 12/2 separate the fractions 2x/2 - 6/2 = 12/2 simplify x - 3 = 6 add 3 to each side x - 3 + 3 = 6 + 3 x = 9 Personally, I think that the first method is easier, since we don't need to worry about separating the fractions. It's also the method that follows the rule the best, and first gets rid of the least connected number (the 6).

Simplify: 3mn + -2n² + 4m - 8mn - 7m

3mn + -2n² + 4m - 8mn - 7m -5mn+ -2n² - 3m

Solve for x in the following equation: 3x + 2 = 3x - 8

3x + 2 = 3x - 8 Subtract 3x from both sides to kick off this algebra party. 3x + 2 - 3x = 3x - 8 - 3x 2 = -8 Uh, 2 can't equal -8 (at least not in this dimension) so the solution is... no solution.

Quadnomial Expression

4 Terms Example: xy - 2x + 3y - 1

the total of forty and a number

40 + x

Solve for x in the following equation: 4x + 10 = 2(2x + 5)

4x + 10 = 2(2x + 5) One step at a time, y'all. Distribute the 2. 4x + 10 = 4x + 10 Subtract 4x. 4x + 10 - 4x = 4x + 10 - 4x 10 = 10 Since 10 always equals 10, this equation is true for all real numbers.

Use the distributive property to simplify 5(6y + -1)

5(6y) + 5(-1) 30y + -5

the product of a number and seven

7x

For what values of x is the following expression undefined? 2x + 5/7x - 6

A fraction is undefined when the denominator equals 0, so we can set the denominator equal to 0 and then solve for x. 7x - 6 = 0 Now add 6 to both sides. 7x - 6 + 6 = 0 + 6 7x = 6 When we divide by 7, we see that x = 6/7 makes this expression is undefined.

6.15

Dividing By Zero

6.9

Dividing Polynomials

How to Undo Multiplication

Division 3x = 21 (Divide both sides by 3.) 3x/3 = 21/3 x = 7

6.17

Equations and Word Problems

Solve for x: 6 - 3(x - 2) = x

First distribute the 3 6 - 3(x - 2) = x 6 - 3x + 6 = x 12 - 3x = x 12 - 3x + 3x = x + 3x 12 = 4x 12/4 = 4x/4 3 = x

Solve for c: (c - 7)/4 = -1

First multiply each side by 4. (c - 7)/4 = -1 [(c - 7)/4]4 = (-1)(4) c - 7 = -4 c - 7 + 7 = -4 + 7 c = 3

Divide 125x²y by 150xy²

For simplicity, we can write this as a fraction: 125xxy/150xyy Now let's write the variables the long way. 125xxy/150xyy Then reduce: Simplified, it look like this: (5*1*1x)/6y = 5x/6y

6.21

Graphing "x" "y" Points

6.23

Graphing Horizontal and Vertical Lines

However, there's one really important rule:

If we multiply or divide by a negative number, we need to flip the inequality sign.

What about the slope of a horizontal line?

Plop these points into the slope equation. m = 2 - 2/-1 - 3 = 0/-4 = 0 We can have a zero as a numerator, which makes the value of the fraction 0. So the slope of a horizontal line is 0, which makes sense because it has no steepness at all.

In this example, y goes up by 3 each time x goes up by 1, so the slope is (prove it by plugging two points into the slope formula). The line intersects the y-axis at 0, so the y-intercept is 0.

Plugging these into our slope-intercept formula, we can come up with an equation. Remember, m is the slope and b is the y-intercept. y = mx + b y = 3x + 0 y = 3x

Graph the equation: y = -x + 2

Remember that -x has a coefficient of -1 We'll start by plotting the y-intercept of 2 and then counting over to the next point using a slope of -1/1.

6.24

Slope

How to Undo Addition

Subtraction or Add the Opposite x + 1 = 5 (Subtract 1 from both sides.) x + 1 - 1 = 5 - 1 x = 4

If a point has a y-coordinate of zero, which axis does it lie on?

The x-axis

What quadrant of the coordinate plane would the point (3, -7) be in?

To get to (3, -7), we go right 3 units on the x-axis and down 7 units on the y-axis. Preferably in a tiny little car. This puts us in the lower-right quadrant, which is Quadrant IV.

6.16

Translating Expressions and Equations

Solve for g: 10 - 2g = 0

Try adding 2g to each side. 10 - 2g = 0 10 - 2g + 2g = 0 + 2g 10 = 2g 10/2 = 2g/2 5 = g

Multiply (x-5)(x+5)

We'll use the box method this time. First, change the subtraction symbol to addition. (x + -5)(x + 5) Now box it up. Add all those terms in the boxes. x² + 5x - 5x - 25 Combine like terms and we're dunzo. x² - 25

Inequalities, like equations, can be translated and used to solve problems. We use an inequality for values that aren't the same, or that can only be the same up to a certain amount.

When a problem says "at least" or "no less than," this means the number given is the very smallest it'll go; it can't get any smaller. The remaining value must be something bigger. For example, if a bag of candy has at least 28 pieces, we know it has 28 or more pieces. So the number of pieces (x) is greater than or equal to 28. x ≥ 28

Multiplying a Monomial by a Monomial

When multiplying a monomial by a monomial, we multiply the coefficients together and tack on the variables at the end (usually in alphabetical order). (14a)(2b) = 28ab

Look Out!!!!!!!!

When using slope-intercept form to graph lines, always make sure the equation is solved for y so that the equation is in the form y = mx + b.

Solving Systems of Linear Equations by Addition

When we have two equations, we can add them together to get a new equation. We don't even need to ask for their permission. Let's say we're trying to solve the following system:

What values of x will make 8/x undefined?

When x = 0, then the denominator will equal 0. That'll make the fraction undefined.

In which quadrant(s) do both the x- and y-coordinates have the same sign?

Where are they both positive or both negative? Quadrant I and Quadrant III

Solve for x: 2x + 6x - 2 = -34

With this equation, we've got two terms with variables. Luckily, they're the same variable, so we can combine the like terms. 2x + 6x - 2 = -34 combine the x terms 8x - 2 = -34 add 2 to each side 8x - 2 + 2 = -34 + 2 simplify 8x = -32 divide each side by 8 8x/8 = -32/8 simplify once more x = -4 Check please! 2(-4) + 6(-4) - 2 = -34 -8 - 24 - 2 = -34 -32 - 2 = -34 -34 = -34

Find the slope of this line: (-4, 1) (0, 2)

You can use any two points that are on the line. However they give us two convenient ones, so let's use them. m = y₁ - y₂/x₁ - x₂ m = 2 - 1/0 - (-4) m = 1/4

Which verbal expression is equivalent to 2(x - 4)? a. The product of two and four less than a number b. The product of two and a number less than four c. Two times a number less than four d. Two times four less than a number e. Four less than two times a number

a. The product of two and four less than a number

What is the equation of a line that passes through (1.5, 1)? a. y = -3/2x + 3/2 b. y = 3/2x + 3/2 c. y = -3/2x - 3/2 d. y = 3/2x -3/2 e. y = 2/3x - 2/3

a. y = -3/2x + 3/2

Solve the equation 2/3 - x = 1/6 for x. a. 1/6 b. 1/3 c. 0.5 d. 5/6 e. 0.75

c. 0.5

Solve this system of equations. x = -3 y = 3x - 6 a. (-3, -15) b. (-3, 15) c. (3, 3) d. (-3, -3) e. (15, 3)

e. (15, 3)

Which algebraic expression is equivalent to the quotient of ten and twice a number? a. 20x b. 8 + 2x c. 10 + 2x d. 10 - 2x e. 10/2x

e. 10/2x

Find the next number in the pattern: 2, 5, 10, 17, 26 ... a. 33 b. 34 c. 35 d. 36 e. 37

e. 37

Common Words and Phrases for: Equals

is

Slope is calculated as change in the vertical direction (y) ÷ change in the horizontal direction (x). This is often called rise over run.

m = change in y/change in x = rise/run

Here's what those extra letters mean:

m is the slope of the line. b is the y-intercept.

Common Words and Phrases for: Add

plus add sum more than in addition to greater than total and

Any number Any given number Some number

x variable Three plus some number 3 + x

a number increased by twelve

x + 12

A number plus two is eight.

x + 2 = 8

The difference between a number and seven is negative three.

x - 7 = -3

Which of the following values of x make the equation x² - 8x = -12 true? x = 1, 2, 3, 4, 5, 6

x² - 8x = -12 x = 1 1² - 8(1) = ? 1 - 8 = -8 x = 2 2² - 8(2) = ? 4 - 16 = -12 x = 3 3² - 8(3) = ? 9 - 24 = -15 x = 4 4² - 8(4) = ? 16 - 32 = =16 x = 5 5² - 8(5) = ? 25 - 40 = =15 x = 6 6² - 8(6) = ? 36 - 48 = =12 The x-values 2 and 6 work.

If the equation of a line is in slope-intercept form, it looks like this:

y = mx + b

Trinomial Expression

3 Terms Example: xy - 2x + 3y

In this phrase, "three times..." means 3 is being multiplied by something. We can write the 3 and an empty set of parentheses waiting for whatever 3 is being multiplied by.

3( ) Next is "the sum of," so we know we'll be adding something inside the parentheses. 3( + ) Finally, "a number and ten" tells us we're adding a variable and ten. 3(x + 10)

How to Undo Subtraction

Addition x - 2 = -7 (Add 2 to both sides.) x - 2 + 2 = -7 + 2 x = -5

6.2

Algebraic Expressions

Do the points (-8, -1), (-6, 1), (-4, 3) and (0, 7) lie in a straight line?

If you don't know, try plotting these points. Yep, they do.

This section will show a few equations where that rule won't work. For problems like these (and any math equation, really), we're going to get rid of the numbers furthest from the x first, and then move in on the x, eliminating numbers as we go. To solve for x in these equations, we need to start with the last operation that we'd do if we were to plug in a value for x.

In the equation (5x + 2)/-3 = 6, we have three numbers connected to x on the left side of the equal sign: x is being multiplied by 5, added to 2, and then divided by -3. To undo all of that mumbo-jumbo, we need to work it backwards.

y = -9x

In the equation y = -9x, the slope is -9 and the y-intercept is 0, since there's no constant. Remember: the slope (m) is equal to the change in y divided by the change in x, or "rise over run."

y = 4 - 8x

In the equation y = 4 - 8x, the slope is -8 (the coefficient of the x-term) and the y-intercept is 4. Don't let the order of the terms trip you up: we can rearrange it so it looks like y = -8x + 4.

Solve for y: 3y + 2 = 5(y - 6)

In this equation we need to first distribute the 5 and then get the y's on one side. They love their privacy. 3y + 2 = 5(y - 6) distribute the 5 3y + 2 = 5y - 30 subtract 5 from each side 3y + 2 - 5y = 5y - 30 - 5y combine the like terms 2 - 2y = -30 subtract 2 from each side 2 - 2y - 2 = -30 - 2 simplify -2y = -32 divide each side by -2 -2y/-2 = -32/-2 Give yourself a high five! y = 16 Now we check it: 3(16) + 2 = 5(16 - 6) 48 + 2 = 5(10) 50 = 50

6.27

Linear Relationships

How to Undo Division

Multiplication x/-2 = 10 (Multiply both sides by -2.) (x/-2)x -2 = 10 x -2 x = -20

6.8

Multiplying Binomials

6.26

Slope-Intercept Form

6.14

Solving Funky Equations

Graph the point (-4, 3)

The first number in the parentheses, a.k.a. the x-value, is -4. This is how far we go horizontally, on the x-axis, from (0, 0). Because it's negative, we go left. We don't put a point there yet, though. We still need to move in the vertical direction to get to the final spot. The second number, or y-value, is 3, so we need to move 3 spaces up. Now we can put the point down.

Coefficients

The number that's multiplied by the variable.

In algebra, we solve equations for the missing variable.

The trick is to keep the scales balanced during all steps. Let's start by looking at a simple example: x + 4 = 6 We know that you know that we know you know the answer (2), but for argument's sake, let's use our scales to solve this. In order to solve for x, we must isolate the variable, or get it all by its lonesome self. To do this, we'd better get rid of that pesky 4. If we only subtract 4 from the left side, the scale will be unbalanced. x + 4 = 6 x + 4 - 4 = 6 - 4 To counter this, we must also subtract 4 from the right-hand side of the equation. Now the scales are balanced once again, and all is right with the universe. x + 4 = 6 x + 4 - 4 = 6 - 4 x = 2 Don't worry; we really don't expect you to draw scales each time you need to solve an equation. We're just using this to illustrate a very important point: we must keep algebraic equations balanced at all times. In order to do this, whatever we do to one side of the equation must be done to the other.

When the two axes meet, they form four quadrants.

These are labeled as Quadrants I, II, III, and IV (usually shown in Roman numerals) and are ordered counterclockwise starting from the upper-right quadrant.

Long ago, and in a guide far, far away, we learned the properties of numbers: commutative, associative, distributive, inverse, and identity.

These properties also apply to adding and multiplying with variables, and they even have the same names.

Solve this system of equations: x = 5 y = -1

This one is pretty chill. If you really think about it, you won't even need to graph it. As you can see, they meet at the point (5, -1). Not a big shocker.

Solve for w: 6w - 9 + 11w = 127

Those like terms are just itching to be combined. 6w - 9 + 11w = 127 17w - 9 = 127 17w - 9 + 9 = 127 + 9 17w = 136 17w/17 = 136/17 w = 8

Look Out!!!!

Watch your negative signs in a fraction bar. 2/-a is the same as -2/a, which is also the same as -(2/a), but it's not the same as -2/-a , which would equal 2/a .

a = b c = d

We can add the left-hand sides of the equations and the right-hand sides of the equations to get: a + c = b + d

Look Out!!!

We can only combine terms with the exact same variables with the same exponents!

The sum of a number and seven is equal to forty-five. What is the number?

We first grab our handy Math-English Dictionary and translate the problem into numeric form. We know that we have a sum, so we're adding two things. Whip out a plus sign. + We're adding some unknown number and seven. Let's use the variable x for the mystery number, then add seven to it. x + 7 The whole deal equals forty-five. x + 7 = 45 Now we've got an equation, so we're ready to solve for x. Subtract 7 from both sides. x + 7 - 7 = 45 - 7 x = 38 You've been owned, equation.

Dividing Polynomials by Monomials

We may also need to divide polynomials by monomials. To do this, we need to separate the "fractions" into smaller fractions with just one term in each numerator. (16x² + 24x)/(-4x²) We can rewrite this fraction as: 16x²/-4x² + 24x/-4x² (Remember, when you add fractions together, you combine the numerators and keep the denominator.) Now, let's write out the variables the long way: (16xx)/(-4xxx)+(24x)/(-4xx) Then reduce: Simplified, it looks like this: (4*1*1)/(-1*1*1) + (6*1)/(-1*1*1) 4/-1 + 6/-x -4 + 6/-x

Distribute -2xy(3x - z)

We need to distribute -2xy to 3x, but what about the z? How do we deal with two negative signs? Does that make the universe implode? Nah, we can handle it. The universe is safe. Just distribute (or multiply) -2xy to 3x and to z, and leave a subtraction symbol between them. (-2xy)(3x) - (-2xy)(z) Now multiply everything. Remember, two negatives make a positive. -6x²y + 2xyz

Look Out!!!!!!

We only switch the inequality sign if we multiply or divide by a negative number. We don't switch it if we add or subtract a negative number.

What if the equation isn't solved for y already?

Well, we'll just have to solve it for y, won't we ? Graph 2x + 3y = 18. First things first: solve for y by subtracting the 2x term and dividing by 3. 2x + 3y = 18 2x + 27 - 2x = 18 - 2x 3y = -2x + 18 3y/3 = -2x/3 + 18/3 y = -2/3x + 6 Now our equation is in slope-intercept form and we can graph it. We put a point at 6 on the y-axis since 6 is the y-intercept. From there, we "rise" -2, which means we go down 2 units, but we don't put a point there yet. We still need to "run" 3 to the right. Then we put the point down.

Simplify the expression -3x² + 2xy - (-y) - 6y² + 10xy + (-2x) - 9x² + y² + 7x - 6y. a. -12x² + 12xy - 5y² + 5x - 5y b. 12x² - 12xy + 5y² - 5x + 5y c. -12x² + 12xy - 5y² + 5x - 7y d. -12x² + 12xy - 7y² + 9x - 7y e. -12x² - 12xy - 7y² - 9x - 7y

a. -12x² + 12xy - 5y² + 5x - 5y

A number decreased by four is equal to three times the same number. What is it? a. -2 b. -1 c. 0 d. 1 e. 2

a. -2

Solve the equation 5x + 3x + 20 = -12 for x. a. -4 b. -1 c. 0 d. 1 e. 4

a. -4

Multiply the expression 5(3x + 6)(3x - 6). a. 9x² - 36 b. 9x² + 36 c. 45x² - 180 d. 45x² + 60x - 180 e. 45x² + 60x + 180

c. 45x² - 180

One square table can fit four chairs around it. Two tables pushed together can fit 6 chairs, and three tables pushed together can fit 8 chairs. How many tables are needed to sit 14 people? a. 4 b. 5 c. 6 d. 7 e. 8

c. 6

Multiply the expression (4z + 2)(z² - 7z + 1). a. 4z³ - 28z² + 4z b. 4z³ + 28z² + 18z - 2 c. 4z³ - 30z² - 18z + 2 d. 4z³ - 26z² - 10z + 2 e. 4z³ - 30z² - 10z + 2

d. 4z³ - 26z² - 10z + 2

Solve the equation 8/x - 1 = 2 for x. a. 2 b. 3 c. 4 d. 5 e. 6

d. 5

What is the equation of a line with points at (-2, 3) (0, -1)? a. 2x + y = -1 b. y = -2x - 1 c. y = -x + 2 d. Both A and B e. None of the above

d. Both A and B

We can also find the slope using a formula. Here's that formula.

m = y₁ - y₂/x₁ - x₂

Find the slope of this line: (0, -0.5)(1, -3.5)

m = y₁ - y₂/x₁ - x₂ m = -0.5 - (-3.5)/0 - 1 m =3/-1 m = -3

Find the slope of this line: (-4, -2) (2, -2)

m = y₁ - y₂/x₁ - x₂ m = -2 - (-2)/-4 - 2 m = 0/-6 m = 0 Since there is no change in the vertical direction, all horizontal lines have a slope of 0.

Find the slope of this line: (2, 2) (2, -2)

m = y₁ - y₂/x₁ - x₂ m = 2 - (-2)/2 - 2 m = 4/0 m = undefined Vertical lines have no change in the x-values, making the denominator zero. We can't divide by zero*, ever, so vertical lines have no slope. This is called an undefined slope. *Go on, try it. Plug 4 ÷ 0 in your calculator...what do you get?

Graph the following inequality: -5 ≤ p < 5

p can be anything greater than or equal to -5 and less than 5. This means that it can only be the stuff in between.

Common Words and Phrases for: Multiply

product of times twice (×2) factor

Graph the following inequality: 3 < x or x ≤ 4

x can be anything greater than 3 or less than or equal to 4. This includes all numbers on the number line, so we need to shade the entire thing!

the quotient of a number and four

x ÷ 4

a number distributed evenly among six

x ÷ 6

What are the x- and y-intercepts of a line that crosses the x-axis at -0.5 and the y-axis at -1?

x-intercept: (-0.5, 0) y-intercept: (0, -1)

What are the x- and y-intercepts of a line that crosses the x-axis at -8 and the y-axis at 2?

x-intercept: (-8, 0) y-intercept: (0, 2)

Graph the equation: y - ¼x = -1

y - 1/4x = -1 add 1/4x to both sides y - 1/4x + 1/4x = -1 + 1/4x simplify y = -1 + 1/4 x switch the -1 and 1/4x and keep the appropriate sign y = -1/4 x + 1 We'll start by plotting the y-intercept of 1 and then counting over to the next point using a slope of -1/4.

Simplify (-21x⁵)/(-7x).

(-21x⁵)/(-7x) = -21xxxxx/-7x = -21x⁴/-7 = 3x⁴/1 = 3x⁴

Simplify (4x²y²z² + 5xyz)/(20x³y³z³)

(4x²y²z² + 5xyz)/(20x³y³z³) Whew, that's a mean-lookin' one. Start by splitting up the fraction: 4x²y²z²/20x³y³z³ + 5xyz/20x³y³z³ = 4xxyyzz/20xxxyyyzzz + 5xyz/20xxxyyyzzz = 1/5xyz + 1/4x²y²z²

Multiply (6a + b)(6a - b)

(6a + b)(6a - b) (6a)(6a) (6a)(-1b) (1b)(6a) (1b)(-1b) 36a² - 6ab + 6ab - 1b² 36a² - b²

A rectangular lawn 12 ft by 15 ft is going to be increased by a uniform amount (x) on each side. What will the new area be?

(x + 12)(x + 15) (x)(x) (x)(15) (12)(x) (12)(15) x² + 15x + 12x + 180 x² + 27x + 180

The general steps we're using here are similar to the ones we used with the substitution method.

1. Eliminate a variable to get an equation in one variable. 2. Solve the equation from step 1. 3. Use one of the original equations to find the value of the other variable. 4. Check your answer in both original equations. So far, the first step ("Eliminate a variable'') hasn't been too difficult. We've only needed to multiply one equation in the system by a number in order to eliminate a variable. Now it's time to up the ante. Of course, we don't mean "ante" in the poker sense. We can't condone gambling, unless it's on the stock market. We frequently need to multiply both equations by something in order to eliminate a variable.

Solve the system of equations: x + 4y = 7 2y - 3x = 8

1. Solve one equation for one variable. The first equation has x all by itself (with a coefficient of 1), so it's easiest to solve that equation for x. Here we go: x + 4y = 7 x + 4y - 4y = 7 - 4y x = 7 - 4y 2. In the other equation, perform substitution to get rid of the variable we solved for in step 1. The other equation is 2y - 3x = 8. Performing substitution gives us 2y - 3(7 - 4y) = 8. 3. After substituting, solve the other equation. We need to solve 2y - 3(7 - 4y ) = 8. Simplify that thing to find 2y - 21 + 12y = 8. Simplify a bit more to get 14y = 29, and divide by 14 to track down y: y = 29/14 Ugh, we're left with a fraction. However, it's the best we can do in this instance. Let's try to overlook our dislike of fractions, though, and make the most of a bad situation. Where are you from, fraction? Oh, really? Well did you...okay, we can't do this. We tried. 4. Find the value of the variable we solved for in step 1. We know that x = 7 - 4y, so plug in y = 29/14 x = 7 - 4 (29/14) = 49/7 - 58/7 = -9/7 Another fraction. A negative one this time. Oh joy! 5. Check that the answer works in both original equations. We think the answer is (-9/7, 29/14 Oy, we almost hope we're wrong. Do these values work in the equation x + 4y = 7? When x = -9/7 and y = 29/14, the left-hand side of the equation is -9/7 + 4(29/14) = -9/7 + 58/7 = 49/7 which is indeed 7. Do these values work in the equation 2y - 3x = 8? Let's see if 2y - 3x really does equal 8 for these bizarro values of x and y. 2(29/14) - 3(-9/7) = 29/7 + 27/7 = 56/7 = 8 How about that; it actually worked! There may be a place for fractions in the universe after all. We were right. The answer is (-9/7, 29/14) So far, each of the systems we've solved using substitution has had exactly one answer, but a system of equations could have no solutions or infinitely many solutions. How's that for a wide range of options? Just somewhere between "none" and "infinity," that's all.

Divide 12a³ + 4a² - 6a by 2a.

12a³ + 4a² - 6a/2a = 12a³/2a + 4a²/2a - 6a/2a = 12aaa/2a + 4aa/2a - 6a/2a = 12a²/2 + 4a/2 - 6a/2a = 6a²/1 + 2a/1 - 3/1 = 6a² + 2a - 3

Solve for y: 15 - (-y) = 20

15 - (-y) = 20 Two negatives? Pshaw. Change 15 - (-y) to 15 + y 15 + y = 20 (subtract 15 from each side) 15 + y - 15 = 20 - 15 y = 5 Don't forget to check your work! 15 - (-5) = 20 20 = 20

Solve for y: 15 ≥ (5y - 10)/2

15 ≥ (5y - 10)/2 15(2) ≥ ((5y - 10)/2)(2) 30 ≥ 5y - 10 30 + 10 ≥ 5y -10 +10 40 ≥ 5y 40/5 ≥ 5y/5 8 ≥ y

Solve for x: 2/3x = 1/2

2/3x = 1/2 Try dividing by 2/3, or multiplying by the reciprocal 3/2. Same diff. (2/3x)(3/2) = (1/2)(3/2) x = 3/4 Don't forget to check your work! 2/3x = 1/2 (2/3)(3/4) = 1/2 1/2 = 1/2

Solve for x: 21x = -147

21x = -147 (divide each side by 21) Go on, give it a go. 21x/21 = -147/21 x = -7 Don't forget to check your work ! 21(-7) = -147 -147 = -147

Method #1

2x - 6 = 12 add 6 to each side 2x - 6 + 6 = 12 + 6 2x = 18 divide each side by 2 2x/2 = 18/2 x = 9

Method #1: Get all the variables on the right-hand side.

2x - 6 = 5x + 18 subtract 2x from each side 2x - 6 - 2x = 5x + 18 - 2x combine the like terms -6 = 3x + 18 subtract 18 from each side -6 - 18 = 3x + 18 - 18 simplify -24 = 3x divide each side by 3 -24/3 = 3x/3 and done! -8 = x

Method #2: Get all the variables on the left-hand side.

2x - 6 = 5x + 18 subtract 5x from each side 2x - 6 - 5x = 5x + 18 - 5x combine the like terms -3x - 6 = 18 add 6 to each side -3x - 6 + 6 = 18 + 6 simplify -3x = 24 divide each side by -3 -3x/-3 = 24/-3 same answer x = -8

three times the total of a number and five

3(x + 5)

This expression can be used in an equation too. Seymour is 5 feet 6 inches tall, and he wants to know how many days it'll be before Audrey II is as tall as he is. We can set this expression equal to Seymour's height (5 feet + 6 inches) and then solve for x.

3x + 10 = 5 feet + 6 inches First we need to get all our numbers in inches. Since Seymour is 5 feet 6 inches, he's 60 inches + 6 inches = 66 inches tall. Now, we solve the equation. 3x + 10 = 66 Subtract 10 from both sides. Then divide both sides by 3. Boom, we found out that , or about 18.7 days. So in about 18 and a half days, Audrey II will be taller than Seymour. Run Seymour, run!

All Real Numbers Solve 3x + 24 = 3(x + 8) for x

3x + 24 = 3(x + 8) distribute the 3 3x + 24 = 3x + 24 subtract 24 from each side 3x + 24 - 24 = 3x + 24 - 24 simplify 3x = 3x divide each side by 3 3x/3 = 3x/3 well duh!!! x = x This means that any number we choose for x will make the equation true. We should verify that this is the correct answer by doing just that: picking a few different numbers and seeing if they work.

five greater than three times a number

3x + 5

No Solution Solve 5 - 6y = 2(-3y) + 1 for y.

5 - 6y = 2(-3y) + 1 First, get rid of those parentheses. Multiply 2(-3) 5 - 6y = -6y + 1 add 6y to each side 5 - 6y + 6y = -6y + 1 + 6y simplify 5 = 1 Wait ?!? 5 ≠ 1 No Solution This equation doesn't work. Since 5 ≠ 1, there is no number we can substitute for y to make this equation true. Unfortunately this one is harder to verify, since it would be impossible to check that every number in the universe does not work. The best way to make sure the answer is correct is to redo the problem.

Graph the line 5x + 10y = 25

5x + 10y = 25 5x + 10y - 10y = 25 - 10y 5x = 25 -10y 5x/5 = 25/5 - 10y/5 x = 5 - 2y y = 0 x = 5 - 2(0) x = 5 - 0 x = 5 (5, 0) y = 1 x = 5 - 2(1) x = 5 - 2 x = 3 (3, 1) y = 2 x = 5 - 2(2) x = 5 - 4 x = 1 (1, 2) Plot and connect. Did you get a straight line?

Which of these equations represents a linear relationship? a. y = x + 7 b. y = (x + 2)(x - 4) c. x = y³ d. y = x + z²

A linear equation has two variables and neither has an exponent. Them's the rules. Answers a and b both appear to have no exponents, but when answer b is distributed, it ends up having an x² term. The answer is a.

Notice that if we could walk the line from the left to the right, we'd be walking uphill. This means that the line has a positive slope. Lo and behold, the slope is positive 1/4.

A positive slope tells us that the line goes uphill, from left to right. A negative slope tells us that the line goes downhill, from left to right.

We can find the constant of proportionality from the table by dividing any y-value by its corresponding x-value (except 0, of course). We can find it from the graph by counting the between two points. We can find it in the equation by finding the slope, 'cause it's the same number.

A proportional relationship can be written as an equation. y = kx

Find a pattern, then fill in the next two numbers: 0, 1, 5, 14, 30, ____, ____

Add the next square number: + 1², + 2², + 3², + 4². The next two numbers are 55 and 91.

Inverse Property

Addition: x + -x = 0 Multiplication: x(1/x) = 1 The Inverse Property states that a number added to or multiplied by its inverse equals the identity. This works for variables too. When we add a variable to the same variable with the opposite sign, we get zero (the additive identity).

Identity Property

Addition: x + 0 = x Multiplication: x(1) = x The Identity Property: The identity for addition, or the additive identity, is 0. This is the number that we can add anything to and it won't change. For example: x + 0 = x. The multiplicative identity is 1. This is the number that we can multiply anything by and it won't change. For example: (x)(1) = x.

Multiply and simplify -4z(6 + 5z) - 3z

Again we need to distribute. Don't forget to stretch first. -4z(6 + 5z) + -3z -4z(6) + -4z(5z) + -3z Now we're cookin'. -4z(6) + -4z(5z) + -3z = -24z + -20z² + -3z Now, don't forget that we need to combine like terms. -24z + -20z² + -3z = -27z - 20z² Time for a victory dance.

Graph the equation 2y = x

Again, we need to solve for y, since it's not in y = mx + b form. 2y = x divide each side by 2 2y/2 = x/2 simplify y = x/2 y = 1x/2 or y = 1/2x Now we can see that the slope of the equation is ½, but where is b, the y-intercept? Can't find it? That's because it's just 0; it's not necessary to write y = ½x + 0. Let's plot a y-intercept of 0 and use the slope of ½ to find a few other points.

Ahmed buys some shoes online for $32.00 per pair including tax, and the shipping is $4.00. If he decides to buy more pairs of those same shoes and it doesn't change the shipping cost, how many can he buy with $100.00?

Ahmed is gonna buy x pairs of shoes, each costing $32.00, so we can write that as a product. 32x Adding on shipping costs, we get: 32x + 4 And this needs to total $100.00. 32x + 4 = 100 We subtract 4 from both sides and then divide by 32. 32x + 4 - 4 = 100 - 4 32x = 96 32x/32 = 96/32 x = 3 Nice. He can buy 3 pairs of shoes.

6.5

Algebraic Properties

What's the equation of a horizontal line passing though the y-axis at 24?

All horizontal lines look like y = (some number) when we flip 'em into equation mode, so we know we won't have any x's in our equation. What number should we set y equal to? That's the million-dollar question. Because it crosses the y-axis at 24, it also has the number 24 in it. So the equation is y = 24. Oh, and we lied about the whole million-dollar thing. Sorry.

Vertical lines are all in x = B form, where B is any real number.

All vertical lines are in the form x = something. Not surprisingly, this is because all points that lie on a vertical line have the same x-coordinate.

When solving systems of equations that have fractions in them, it's best to check the answers in the original equations.

Although it's tempting to check the answers in the nicer equations, what if we made a mistake when getting rid of the fractions? Almost inconceivable, we know, yet possible. Then we'd be finding the right solutions for the wrong equations, which wouldn't help us any more than if we were to be in the right place at the wrong time. For example, at the Kodak Theatre four months before the Oscars, or in Central Park at 4 a.m.

Look Out!!!!!!!

Although we can graph a line by only plotting two points, it's always a good idea to do at least three. If all three lie in a straight line, we can feel pretty confident that our answer is correct.

Point-Slope Form

Another form of a linear equation that's super useful when we only know a few points is the point-slope form, which looks like this. (y - y1) = m(x - x1)

6.1

Arithmetic, Geometric, and Exponential Patterns

Look Out!!!!!

Be sure that you're solving for the variable, not the opposite of the variable (-x).

Look Out !!!!!!

Be sure to write your answer as a point in (x, y) form.

Look Out!!

Be very careful with "less than." Three less than a number is translated as "x - 3." The reverse of that, "3 - x," would be a number less than 3.

Which of the following is the best translation of the numeric expression 20(a + 6) - 18? a. Twenty less than eighteen times the sum of a number and six. b. Eighteen less than the sum of a number and six. c. The product of twenty and the sum of a number and six. d. Eighteen less than the product of twenty and the sum of a number and six.

Because 18 is being subtracted from the expression, the phrase "eighteen less than" needs to come at the beginning of the translation. Bam, that immediately eliminates answers a and c. The expression begins with a product because we're multiplying 20 by something. So after the phrase "eighteen less than," we should have a product. That eliminates answer b. Answer d is left, and it includes "eighteen less than" and "the product of twenty..." and "the sum of a number and six." It includes all three operations, and in the right order. We have a winner.

Solve for y: -14/(y-2) = -10

Bleh, that annoying variable is on the bottom again. We need to multiply each side by (y - 2) in order to get it out of the denominator. -14/(y-2) = -10 multiply by (y-2) [-14/(y-2)](y-2) = -10(y-2) the (y-2)'s on the left cancel -14 = -10(y-2) distribute the -10 -14 = -10y +20 subtract 20 from each side -14 - 20 = -10y + 20 - 20 simplify -34 = -10y divide each side by -10 -34 ÷ (-10) = -10y ÷ (-10) 3.4=y now we're talkin' And check: -14/(3.4 - 2) = -10 -14/1.4 = -10 -10 = -10

Let's go through an example very carefully: 4(3x +z)

By applying the distributive property, we can multiply each term inside the parentheses by 4. This is called "distributing." 4(3x) +4(1) = 12x + 4 Since 12x and 4 are not like terms, this is as far as we can go with the problem. What about subtraction? Let's look at a subtraction problem using two different methods. -2(7-2x)

Compound Inequalities

Compound inequalities are two or more inequalities combined in the same statement. They often include the words "and" or "or." With "and" inequalities, we only graph the numbers that satisfy both inequalities, a.k.a. the intersection of both inequalities. With "or" inequalities, we graph the numbers that satisfy either inequality, or both at the same time. In other words, we graph the combination, or union, of both inequalities.

Method 3: Box Method

Create a two-by-two table. Place one factor on top, and the other on the side. It doesn't matter which goes where, since multiplication is commutative. Be sure to keep the subtraction and addition signs with the correct terms. Now, multiply each factor in the rows with each factor in the columns and write the products in the boxes. Add them all together. 2x² + x - 6x - 3 Combine like terms. 2x² - 5x - 3 And we're done! Yup: same answer again. Use whichever method you like best.

Dividing Polynomials

Dividing polynomials starts with dividing monomials, and dividing monomials boils down to reducing fractions, and reducing fractions? Pshaw, we've been doing that for eons. No big whoop. The fractions have variables now, but so what? We've got this. The most important thing to remember is that when we divide a variable by itself, it equals 1, just like 5 ÷ 5 = 1 or x ÷ x = 1.

If you encounter a variable on both sides of the equal sign...

Don't assume it's a typo and move on to the next problem; it may very well be there on purpose. The key to solving these types of equations is to move all the terms containing the variable to one, and only one, side. It doesn't matter which side you choose. Just try to pick the easier one.

Whatever you do to one side of an equation, you MUST do to the other.

Equations are like carefully balanced scales. Imagine an old-timey scale. If both expressions on each side of the equal sign match, then they're balanced. 3 + 2 = 5 If one side is heavier than the other, the scales are tipped. 3 + 2 > 4

Some phrases tell us the numbers in reverse order. The most common is the phrase "less than," which first tells us what's being subtracted and then tells us what it's being subtracted from. For example, in the phrase "six less than a number," 6 is being subtracted from some given number, so the 6 goes at the end. It looks like this when we translate it into math: x - 6

Finally, some phrases combine several operations into one long expression. We translate in the order everything is given, with the exception of "less than," and it often helps to include parentheses. For example: "Three times the sum of a number and ten."

Solve for z: z + 2 ≤ 6

First multiply each side by -3. (z+2)/-3 ≤ 6 multiply each side by -3 -3((z+2)/-3) ≤ -3(6) switch the sign since we multiplied by -3 z + 2 ≤ -18 subtract 2 from each side z + 2 - 2 ≤ -18 - 2 simplify z ≤ -20 We switched the sign in the second step because we multiplied by a negative number. z ≥ -20 We got z ≥ -20, so check by plugging in any number ≥ -20, like 0. (0+2)/-3 ≤ 6 -2/3 ≤ 6 Yup, -⅔ is smaller than 6!

Simplify 4xyz - 9xym/2xy

First we give the denominator to both terms in the numerator to split the fraction up into two fractions. Sharing is caring, you know? 4xyz/2xy - 9xym/2xy Next, we cancel a bunch of stuff and simplify each fraction. 2z/1 - 9m/2 = 2z - 9m/2 Ah, much better.

We know that we can graph a linear equation with two variables, such as the famous x and y, as a straight line. (By the way: that's true only because the variables don't have any exponents, but that's for another day). There are a few special equations that also produce a straight line. Let's look at this one: y = 3. We can graph it by making a table of points.

First we pick a few random values for x, like -2, 0 , and 3. Then we plug these numbers into the equation in place of x. But wait—there is no x variable. No matter what x is, y will always be 3. Easiest substitution ever. (-2, 3) (0, 3) (3, 3) Now we plot these points on a grid and get a line. If we pick any point on the line, like the three shown, the y-coordinate will be 3. The x-coordinate will vary, but y will always be 3. That's why the equation for this line is y = 3.

Multiply (5y + 3x)(8y - 1)

First, change the subtraction symbol to the addition symbol and make the 1 negative. (5y + 3x)(8y + -1) Now use the distributive property twice, like a champ. (5y + 3x)(8y) + (5y + 3x)(-1) = (5y)(8y) + (3x)(8y) + (5y)(-1) + (3x)(-1) All together now: 40y² + 24xy - 5y - 3x There are no like terms to combine, so we're done.

Solve the system of equations: 1/2x + 2/3y = 1 3x - 2/5y = 2

First, we get rid of the fractions. Multiply the first equation through by 6 and the second equation through by 5 to find (6)1/2x + (6)2/3y = (6)1 3x + 4y = 6 (5)3x - (5)2/5y = (5)2 15x - 2y= 10 We know what to do from here. In this case, we can eliminate either variable without too much trouble, assuming they don't get frisky. Let's eliminate y, since its coefficients already have opposite signs. Thank you, y, for cooperating. Multiply the second equation by 2, so now we're looking at the system of equations 3x + 4y = 6 30x - 4y = 20 We add these equations to find 33x = 26, which means, unfortunately, that x = 26/33 Use the first original equation to find y. We plug in our value of x into 1/2x + 2/3y = 1, which gives us 1/2 (26/33) + 2/3y = 1 This simplifies to 2/3y = 20/33 and then we multiply both sides by the reciprocal of y's coefficient to get y = 20/33 x 3/2 = 10/11 Now we think the solution to the system is (26/33, 10/11), but we need to check our answer in the other equation, which was 3x - 2/5y = 2. Let's plug in the values we found for x and y and see if they work. If they don't, we'll try to keep from throwing a hissy fit. Look at things through a clean pair of perspectacles. 3 (26/33) - 2/5 (10/11) = 26/11 - 4/11 = 22/11 Yep, that's the same thing as 2. These values work, so we found the solution to the system of equations. Yeesh, finally.

Divide (2x²y - 6xy²)/10yx

First, we need to separate this into two fractions: 2x²y/10xy - 6xy²/10xy Now let's write out the variables the long way, without exponents: 2xxy/10xy - 6xyy/10xy Then reduce: Simplify: (1*1*1x)/(5*1*1) - (3*1*1y)/(5*1*1) = x/5 - 3y/5 = (x-3y)/5 In the final step we were able to add the fractions together, since they shared a common denominator.

Divide 15ab + 5a - 10b by 5ab

For easiness' sake, we can write this as a fraction: (15ab + 5a - 10b)/5ab Next, separate the fractions: 15ab/5ab + 5a/5ab + - 10b/5ab Then reduce: Simplify: (3*1*1)/(1*1*1) + (1*1)/(1*1b) - (2*1)/(1*1a) = 3/1 + 1/1b - 2/1a = 3 + 1/b - 2/a This is as simplified as we can get, because the fractions have different denominators.

Sometimes equations have more than one term with a variable. In that case, we need to combine all the like terms and then solve.

For example, in the equation 3x - 2 + 6x = 16, there are two terms with an x: 3x and 6x. We can combine these to get a simpler equation. 3x - 2 + 6x = 16 9x - 2 = 16 Now we can solve for x by adding 2 and dividing by 9. 9x - 2 = 16 9x - 2 + 2 = 16 + 2 9x = 18 x = 2 AND the answer is x = 2.

Linear and Non-Linear Equations

For example: y= 2/3x - 3 is a linear equation because the variables all have an invisible exponent of 1 (which seems like they have no exponent). But the equation y = 3x² + 1 is not a linear equation because the x variable has an exponent of 2. Here are a few more examples. t = v - 5 = linear equation t = v² - 5 = non-linear equation (y - 6) = 4(x + 1) = linear equation (y - 6) = 4(x + 1)³ = non-linear equation 5x - 3y = 2x + 7 = linear equation 5x(x) - 3y = 2x + 7 = non-linear equation

How to Solve Systems of Equations by Graphing

Graph both lines on the same coordinate grid. Since they're both in slope-intercept form, we can do this by plotting the intercepts, then using the slope to find another point. It's pretty clear that these lines meet at the point (2, 2), which is our answer!

6.18

Graphing Inequalities on a Number Line

6.22

Graphing Lines by Plotting Points

Most of the lines we'll be graphing will much more complex than simple vertical and horizontal lines. There are many ways to go about graphing these, but we'll only work with the two most common methods: plotting points and slope-intercept form.

Graphing lines by plotting points isn't too rough. Just find two or more points—any (x, y) points—on the line and connect the dots.

What type of quadrilateral do the points (-3, 1), (4, 1), (2, -2) and (-5, -2) form?

Have you bothered to graph them yet? They form a parallelogram.

Solve for y: 1 -3y = -14

Here we can isolate the variable by first getting rid of the 1. 1 - 3y = -14 add (-1) to each side 1-3y + (-1) = -15 + (-1) simplify -3y = -15 divide each side by (-3) -3y/-3 = -15/-3 y = 5 oh heck yes! Remember to check it! 1 - 3(5) = -14 1 - 15 = -14 -14 = -14

Solve for y: 3y + 2 > 12 - y

Here you can choose which side of the inequality to get the variables on. It's really up to you and whatever way makes more sense. In this problem, we'll move the variables to the left side. 3y + 2 > 12 - y add y to each side 3y + 2 + y > 12 - y + y remember y is the same as 1y 4y + 2 > 12 subtract 2 from each side 4y + 2 - 2 > 12 - 2 simplify 4y > 10 divide each side by 4 4y/4 > 10/4 notice that the sign didn't switch y > 5/2 or 2.5 Since we divided each side by positive 4, we don't need to switch the > to <. To check, we need to pick a number greater than 2.5 and plug it into the original inequality. Let's try 4. 3(4) + 2 > 12 - 4 12 + 2 > 12 - 4 14 > 10 Well, 14 is greater than 4, so we're all good!

Graph the equation y = -0.5x - 2

How about this: we'll pick some values for x and solve for y, since y is already isolated. x = 0 y = -0.5x - 2 y = -0.5(0) - 2 y = 0 - 2 y = -2 (0, -2) x = 1 y = -0.5x - 2 y = -0.5(1) - 2 y = -0.5 - 2 y = -2.5 (1, -2.5) x = 2 y = -0.5x - 2 y = -0.5(2) - 2 y = -1 - 2 y = -3 (2, -3) Now let's plot them coordinates and connect them dots.

The inequality y < 2 means that y can be any number less than 2 (such as 1.9, 0.75, 0, -6, etc.). The inequality y > 7 means that y can be any number greater than 7 (such as 7.1, 8, 9, 537, etc.). The inequality y ≤ 2 means that y can be any number less than 2, or it can be equal to 2 itself (2, 1.9, 1, 0, -6, etc.). Last but not least, the inequality y ≥ 7 means that y can be any number greater than 7, or it can be equal to 7 (7, 7.001, 8, 9, 200, etc.). That little line underneath an inequality symbol means "or equal to."

How do we remember which one is which? "Less than" and "greater than" are easy to mix up, so we like to think of them as an incomplete Pac-Man (or, if you prefer, Ms. Pac-Man). Pac-Man, being the hungry circle he is, always wants to eat the bigger number, so his "mouth" will always be open towards the larger number.

Chances are, we've been graphing points for a long time.

However, we've probably been doing so on charts that look like this:

We need to buy some grass skirts for a luau and each one costs $2.00 (tax included). Write a table, a graph, and an equation to represent this relationship, then determine if it's a proportional relationship.

Hula time. If we buy 1 skirt, it'll cost $2.00. If we buy 2 skirts, it'll cost $4.00. If we buy 3 skirts, it'll cost $6.00. We can put this in a table. Skirts Cost 1 2 2 4 3 6 To graph the line, we can assume the number of skirts is the x variable and the cost is the y variable. Then we graph it. To write an equation, we need the slope of the line and the y-intercept. Use the slope formula and the last two points, (2, 4) and (3, 6). 6 - 4/3 - 2 = 2/1 = 2 Boom, the slope is m = 2. The line crosses the y-axis at 0, so the y-intercept is b = 0. Now plug 'em both into our slope-intercept equation. y = mx + b y = 2x + 0 y = 2x This relationship is proportional because the equation is in the form y = kx where k = 2. (In other words, it has a slope and no intercept.) We can also see that it's proportional because when we divide any y-value by its x-value, we always get the same number, 2, which is the constant of proportionality.

Proportional Linear Relationships

If a king-sized candy bar costs $3.00 and we buy 1, we pay $3.00 (just for giggles, we'll say there's no sales tax). If we buy 2 we pay $6.00, if we buy 3 we pay $9.00, and if we buy 4 we pay $12.00. Let's put this info in a table so we can check it out a bit more. We'll say x is the number of candy bars purchased and y is the amount of money spent. x y 0 0 1 3 2 6 3 9 4 12 They say imitation is the highest form of flattery, so the x variable in this relationship should be very flattered. Each time it increases, so does y, and by the same amount too. This is called a proportional relationship. Whaddaya know, proportional relationships are linear !!!

Solve this system of equations using substitution: y = 2x + 5 2y = 4x + 10

In this system the first equation has the y isolated, so we can plug it into the second equation. 2(2x + 5) = 4x + 10 Simplify that thang. 4x + 10 = 4x + 10 Subtract 4x from both sides and things get a little weird. 10 = 10 This is true everywhere, so there are an infinite number of answers.

6.20

Inequality Word Problems

6.25

Intercepts

Solve for x: 4/(2x-1) ≥ -2

Just like with equations, we need to get the variable out of the denominator. We start by multiplying each side by (2x - 1). 4/2x - 1 ≥ -2 multiply each side by (2x - 1) (4/2x - 1)(2x - 1) ≥ -2(2x - 1) the (2x - 1)'s cancel on the left 4 ≥ -2(2x - 1) distribute the -2 4 ≥ -4x + 2 subtract 2 from each side 4 - 2 ≥ -4x + 2 - 2 simplify 2 ≥ -4x divide each side by -4 2/-4 ≥ -4x/-4 -1/2 ≥ x switch the sign from ≥ to ≤ -1/2 ≤ x Since we divided each side by negative 4, we switched the sign from ≥ to ≤. To check, plug in any number greater than or equal to -½. Since it can equal -½, let's plug that in. 4/(2(-1/2) - 1) ≥ 2 4/(-1-1) ≥ -2 4/-2 ≥ -2 -2 ≥ -2 -2 is greater than or equal to -2!

As before, there are a lot of different ways to solve these problems. Choose whatever way floats your boat.

Let's dive into two different methods for solving the following equation. 2x - 6 = 5x + 18

Solve this system: y = (1/2)x + 1 2y - x = 8

Let's do substitution. All the cool kids are doing it. 1. Solve one equation for one variable. The first equation is already solved for y, which makes our lives better. y = (1/2)x + 1 2. In the other equation, perform substitution to get rid of the variable we solved for in step 1. We substitute for y in the equation 2y - x = 8 to get: 2 ((1/2)x + 1) - x = 8 3. After substituting, solve the other equation. To solve 2 ((1/2)x + 1) - x = 8, first we simplify to find x + 2 - x = 8. Then we run into trouble. Even if "Trouble" is your middle name, you're not going to like what comes next. When we combine the x terms, we're left with the statement 2 = 8. Uh-oh. We know 2 doesn't equal 8, or else the Raptors got royally ripped off by the official scorers at last night's game. What does this incorrect equation tell us? There's no solution to the system. For these lines to intersect, 2 must equal 8, which is ridiculous. Our calculator agrees. Whenever substitution leads us to such a ridiculous and impossible statement, it means the system of equations has no solution, which leads us to an important life lesson: being ridiculous and impossible never solves anything. On the other hand, if substitution leads us to a statement that's always true, such as 1 = 1, it means that the lines are actually the same, and every point on either line is a solution. Everyone is happy, and order is restored. Now the only thing that's ridiculous is how much we're enjoying solving substitution problems. If one or both of the equations in a system contains fractions, we get rid of the fractions and then proceed as usual. We know you like this news. It's actually a little scary how much you enjoy getting rid of fractions; we only hope it doesn't give way to more destructive behavior, such as torturing polynomials.

Slope-intercept form of a line: y = mx + b

Let's examine how to graph an equation in slope-intercept form. What does the graph of y = 2x + 1 look like? Our equation is in slope-intercept form, so we know that the number in front of x is the slope (2), and 1 is the y-intercept. We start by plotting the y-intercept. Next, since we know that the slope is 2, also known as 2/1, we know that another point will be 2 units up and 1 unit over (in the positive direction of course). Finally, we connect these points.

An intercept can be written in (x, y) form or in words, like "the x-intercept is 2." It can't be written as x = 2, because this is an equation of a vertical line going through the point (2, 0).

Let's find the y-intercept of the equation y = 2x + 1. To find the y-intercept, we substitute 0 in for x and then solve for y. y = 2(0) + 1 y = 1 So the y-intercept is where x = 0 and y = 1, or the point (0, 1). To find the x-intercept, we replace y with 0 and then solve for x. First off, subtract 1 from both sides. 0 = 2x + 1 0 - 1 = 2x + 1 - 1 -1 = 2x Now divide both sides by 2. -1/2 = 2x/2 -1/2 = x So the y-intercept is where x = -1/2, or the point (-1/2, 0).

Solve for x: -3x - 7 = -12

Let's first get rid of the 7, since it's not attached to the variable. -3x - 7 = -12 add 7 to each side -3x - 7 + 7 = -12 + 7 simplify -3x = -5 divide each d=side by -3 -3x/-3 = -5/-3 simplify x = 5/3 To check, let's plug that fraction in for x. -3(5/3) - 7 = -12 -5 - 7 = -12 -12 = -12 It works!

Dividing Monomials by Monomials

Let's look at an example. Remember, fractions are just another way to write division. 18y³/9y Instead of writing y^3, we can write yyy (which means y x y x y). 18yyy/9y Now we can divide, or reduce, the coefficients and the variables. 18 / 9 = 2 and y/ y=1, just like 7/7=1. Simplified, this looks like: (2x1yy)/(1x1) = 2y²/1 =2y²

Find the equation of the line in slope-intercept form that passes through (0, 1) and (3, 2)

Let's look carefully at the graph and see if we can find any important information.... Right off the bat, we can see that the y-intercept is 1. We can also find the slope by counting the change in the vertical and horizontal directions between the two points shown. From the point (0, 1) to (3, 2), our line is has a "rise" of 1 and a "run" of 3. Slope is rise over run, so that gives us a slope of ⅓. Now using that information, we know these things: b = 1 and m = ⅓ All we need to do is plug them into our equation, y = mx + b. y = ⅓x + 1

Method 1: Distributing the first factor to both terms in the second factor

Let's say we want to multiply these two binomials together: (x - 3)(2x + 1) First we'll make things easier by changing the subtraction symbol to adding a negative number. (x + -3)(2x + 1) We need to distribute (x + -3) to both terms in the second binomial, to both 2x and 1. If we apply the distributive property once, it looks like this: We're multiplying the entire first binomial by 2x and by 1. (x + -3)(2x) + (x + -3)(1) Now we can run through the distributive property again on each term, like so: (x)(2x) + (-3)(2x) + (x)(1) + (-3)(1) = 2x² + (-6x) + x + (-3) = 2x² - 6x + x - 3 Hey, look at that. We've got two terms with x's in them, so we can combine like terms. 2x² - 6x + x - 3 = 2x² - 5x - 3 Whew, there's our answer. Let's see if we get that same answer using the other methods.

Graph the equation 2y = 6x

Let's solve this one for y 2y = 6x divide both sides by 2 2y/2 = 6x/2 simplify y = 3x Now, let's plug in some numbers for x and solve for y x = 0 y = 3x y = 3(0) y = 0 (0, 0) x = 1 y = 3x y = 3(1) y = 3 (1, 3) x = -1 y = 3x y = 3(-1) y = -3 (-1, -3) Plot and connect.

Let's review a math rule. Remember this one? No dividing by zero. Nuh-uh, no way, never, ever. Don't do it. Ever wonder why? A division problem can be read as a multiplication problem that's missing a number. "What is 24 divided by 6?" is the same question as "6 times what is 24?" Both answers are 4.

Let's try that with 0. The division problem "what is 24 divided by 0?" is the same as the question "0 times what is 24?" And boom, there it is—impossible, no answer, can't be done, nuh-uh, no way, never ever. There is nothing we can multiply 0 by to get 24. In fact, there is nothing we can multiply 0 by to get any number (other than 0 itself), because 0 times anything is just 0. A number divided by 0 is like matter getting sucked into a black hole: it becomes undefined.

Forms of Linear Equations

Linear equations and inequalities are masters of disguise, and they can be written in lots of different forms. Don't let them fool you—they're still linear.

Representations of a Linear Relationship

Linear relationships can be represented in several different ways: a graph, an equation, or a list of points. If we have one of these, with a little sleuth work we can figure out the other ones. We've already started with a linear equation and created a table of points and a graph. Rewind: if we start with a table of points or with a graph, we can go backwards and figure out the equation. If we have a table of points that have a linear relationship, we can use two points to find the slope of the line. We can then graph the line to find the y-intercept, and use the slope and y-intercept to write an equation of the line.

A plant is 13 inches tall and grows 2 inches a week. How many weeks will it take for the plant to be 23 inches tall?

Man, somebody sure has a green thumb. We add the current height of the plant to the product of the rate it's growing per week (2 inches) and the number of weeks it grows (n) and set this equal to 23. Then we solve for n. Translated into math, that gives us the equation: 13 + 2n = 23 Subtracting 13 from both sides, we get: 13 + 2n - 13 = 23 - 13 2n = 10 Divide both sides by 2 and we get n = 5. In 5 weeks the plant will be 23 inches tall. We're gonna need a bigger pot.

Solve for m: m/(m + 7) = 2

Multiply each side by (m + 7) m/(m + 7) = 2 [m/(m + 7)](m + 7) = 2(m + 7) m = 2m + 14 m - 2m = 2m + 14 - 2m -1m = 14 -1m/-1 = 14/-1 m = -14

Solve for x: (37 - 12)/(x+1) = 5

Multiply each side by (x + 1). (37 - 12)/(x + 1) = 5 [(37 - 12)/(x + 1)](x + 1) = 5(x + 1) [25/(x + 1)](x + 1) = 5(x + 1) 25 = 5(x + 1) 25/5 = 5(x + 1)/5 5 = x + 1 5 - 1 = x + 1 - 1 4 = x

Solve for y: -4 = (-4y-8)/(y+2)

Multiply each side by (y + 2) -4 = (-4y - 8)/(y + 2) -4(y + 2) = [(-4y - 8)/(y + 2)](y + 2) -4y - 8 = -4y - 8 -4y - 8 + 8 = =4y - 8 + 8 -4y = -4y -4y/-4 = -4y/-4 y = y

For what values of x is the following expression undefined? 2/x

Newsflash: A fraction is undefined when the denominator equals 0. Dividing by 0 is the biggest math no-no in the book. The denominator is just x, so this expression is undefined when x = 0.

The last operation on the list is division by -3, and the number furthest from the x is -3. So we're going to get rid of that "divide by -3" first by multiplying both sides by -3. (Notice we convert the integer -3 into a fraction to be able to multiply it on the left side of the equation.)

Next, we need to get rid of the "add 2" by subtracting 2 from both sides. 5x + 2 - 2 = -18 - 2 5x = -20 Finally, we divide by 5. 5x/5 = -20/5 x = -4 There's our answer.

If we solve an equation and get 6 = 10 in the very last step, is the answer all real numbers or no solution?

No Solution 6 ≠ 10 There are two different funky answers we can get when solving an equation. When the values are exactly the same, such as 4 = 4 or x = x, the answer is all real numbers. When the values are not equal, like 5 = 9 or -3 = 16, the answer is no solution. The answer to this equation is no solution.

Pick an x-value = 0 Plug it into y = 2(0) + 1 Solve for y: y = 2(0) + 1 y = 0 + 1 y = 1 (x, y) = (0, 1) Pick an x-value = 1 Plug it into y = 2(1) + 1 Solve for y: y = 2(1) + 1 y = 2 + 1 y = 3 (x, y) = (1, 3) Pick an x-value = -1 Plug it into y = 2(-1) + 1 Solve for y: y = 2(-1) + 1 y = -2 + 1 y = -1 (x, y) = (-1, -1)

Now that we have our three points, we can plot these on a coordinate grid and connect 'em.

Constants

Numbers that don't change.

System of Linear Equations

Occasionally we'll be given two linear equations, also known as a system of linear equations, and asked to solve for x and y. There are tons of different ways to solve a system of linear equations, including graphing the lines, substitution, and elimination. However, in pre-algebra we'll only touch on the first: solving by graphing. Solve for x AND y: y = 2x - 2 y = -3/2x + 5 What this question is really asking: find the (x, y) point where these lines meet.

How to Check Your Answer

One of the best things about solving equations is that we can, and should, plug our answer back into the original equation to see if it works. Let's look at that last division problem. x/-2 = 10 (x/-2)(-2) = 10(-2) x = -20 If we plug -20 back into the original equation, both sides should be equal. -20/-2 = 10 10 = 10 Since both sides equal 10, we know that our answer is correct!

Calculating the Slope

One way to find the slope is to pick any two points, then count out how far we need to "rise" to get from the first point to the second point. (x₁, y₁) = (-4, 1) (x₂, y₂) = (4, 3) Using the points above, we need to go up 2 spaces from the first point to be in line with the second point. This becomes our numerator. slope = rise/run = 2/run Then we count how far we need to "run" horizontally to get from the first point to the second point. The second point is 8 units to the right of the first point. This becomes our denominator. slope = rise/run = 2/8 We simplify the fraction and get a slope of m = 1/4.

Which axis does the graph of y = 4 cross through and at what point?

Poor little x. It wasn't invited to the graphing party. Maybe its invitation got lost in the mail? There's no x variable at this party, so this graph crosses through the y-axis. But where? It crosses at y = 4, but we need to name this point. Because our equation slices through the y-axis, the x-value is 0 so the point is (0, 4).

Solve this system of equation using substitution: y = 3x + 5 y = x - 3

Since both equations have y isolated, we can substitute either one into the other. Let's substitute the first equation into the second one. y = 3x + 5 (3x + 5) = x - 3 Now we solve for x: 3x + 5 = x - 3 3x + 5 - x = x - 3 - x 2x + 5 = -3 2x + 5 - 5 = -3 - 5 2x = -8 Dividing by 2, we get x = -4. Now we can substitute -4 into the first equation in place of x. y = 3(-4) + 5 y = -12 + 5 y = -7 So the solution is (-4, -7).

Solve the system of equations: 3y = 1/2x + 1 2x - 1/4y = 1

Since each equation has fractions, we'll get rid of the fractions first. Multiply both sides of the first equation by 2: 6y = x + 2 And multiply both sides of the second equation by 4: 8x - y = 4 Now, instead of solving the original system of equations, we can solve the system 6y = x + 2 8x - y = 4 We know how to do this: we just follow the same steps we've been following. They have to lead somewhere. 1. Solve one equation for one variable. Since x has a coefficient of 1 in the first equation, we'll solve the first equation for x to get: x = 6y - 2 2. In the other equation, perform substitution to get rid of the variable we solved for in step 1. We replace the x in the second equation: 8x - y = 4 8(6y - 2) - y = 4 3. After substituting, solve the other equation. We need to solve the equation 8(6y - 2) - y = 4. Simplify to find 48y - 16 - y = 4, and rearrange to get y = 20/47 4. Find the value of the variable we solved for in step 1. Since we found that x = 6y - 2 when y = 20/47, we get x = 6(20/47) - 2 = 120/47 - 94/47 = 26/47 We're not thrilled about these fractions, but we also weren't thrilled after seeing the previews for Adam Sandler's Jack and Jill, and look how great that turned out! (Psych. It was awful.) 5. Check that the answer works in both original equations. We think the answer is (26/47, 20/47) Let's make sure these values work in the original equations—you know, the ones with the fractions in them. First, let's check that we have a solution to the equation 3y = 1/2x + 1. When we substitute in the values we found for x and y, the left-hand side of this equation is 3 (20/47) = 60/47. The right-hand side of the equation is 1/2 (26/47) + 1 = 13/47 + 47/47 = 60/47. Since the left-hand and right-hand sides of the equation agree, we have a solution. It gives us even more confidence in our solution to know that four out of five dentists also agree. Second, let's check the equation 2x - 1/4y = 1. When we substitute in the values we found for x and y, the left-hand side of this equation is 2 (26/47) - 1/4 (20/47) = 52/47 - 5/47 = 47/47 = 1 which agrees with the right-hand side of the equation. Houston, we also have a solution to the second equation. We can safely say that the solution to the system of equations is (26/47, 20/47).

With this last example, if we had divided by positive 2 instead of -2, we would've found that -x > 1. So, x < -1 and -x > 1 are really the same thing! That's why we need to switch the sign when we divide or multiply by a negative.

Since we divided by -2, we switched the sign from > to <. Now, just like with equations, we can check our answers. Since x < -1, pick any number less than -1 and plug it into the original inequality (we picked -2). -2(-2) + 3 > 5 4 + 3 > 5 7 > 5 Yup, 7 is greater than 5, so we can be pretty confident that we solved this correctly. However, unlike equations, we can't be completely sure. If we want to double check our work, that wouldn't be a horrible idea.

This little fact explains a lot of messes in the world of Algebra. For example, the simple little expression 1/x is easy-peasy if x = 2 or 81 or -2347, but if x = 0, it melts into a puddle and evaporates into nothing.

So now that we have variables (letters that supposedly can be any number) and fractions (two numbers being divided), we need to make sure no zeros sneak into our denominators. If a fraction does have 0 in the denominator, it's undefined.

6.13

Solving Equations with Variables on Both Sides

6.19

Solving Inequalities

6.12

Solving More Complex Equations

6.29

Solving Multiple Equations Algebraically

6.28

Solving Multiple Equations by Graphing

6.10

Solving One-Step Equations

6.11

Solving Two-Step Equations

Solving Inequalities

Solving inequalities isn't that much different than solving equations. Instead of having an equal sign divide the two sides, there's an inequality sign.

Two-Step Equations

Solving two-step equations isn't much more complicated than solving one-step equations; it just involves an extra step. Usually, there's more than one way to solve these. It's okay to use whatever method makes the most sense to you. The general rule of thumb when isolating the variable is to undo the order of operations, PEMDAS. Start with addition and subtraction, then multiplication and division, then exponents, and finally parentheses. Let's look at an example: Solve 2x - 6 = 12 for x.

The slope in this equation (3) is also called the constant of proportionality. Big word, small little number. In a nutshell, the constant of proportionality is the number we multiply x by to get y.

Sometimes it's called a constant rate of change, too. So slope, constant of proportionality, and constant rate of change: all the same thing.

Solving Funky Equations

Sometimes we'll need to solve an equation that has a funky answer, like 10 = 8 or y = y. This doesn't necessarily mean that we did anything wrong; it might very well mean that all or no numbers work. Here are some of these equations.

The phrases "at most" and "no more than" both mean the number given is the biggest the value will ever get. If a box of drinks has no more than 15 drinks, it has 15, or 14, or 13...drinks. Notice these values are all less than or equal to 15. x ≤ 15

Sometimes we're lucky and the problem just uses the phrase "less than" (<) or "greater than" (>). But humans are not robots—we like to get creative about our phrasing—so don't count on that.

Solve for z: -12z + 18 = -18

Start by adding -18 to each side. -12z + 18 = -18 -12z + 18 + -18 = -18 + -18 -12z = -36 -12z/-12 = -36/-12 z = 3

Solve for j: 3(3j + 12) = -1(15 - 9j)

Start by distributing the 3 and -1 3(3j + 12) = -1(15 - 9j) 9j + 36 = -15 + 9j 9j + 36 - 36 = -15 + 9j - 36 9j = -51 + 9j 9j - 9j = -51 + 9j - 9J 0 = -51 0 ≠ -51 No Solution

Solve for k: (-k+8)/15 = 2

Start by multiplying each side by 15. (-k+8)/15 = 2 [(-k+8)/15]15 = 2(15) -k + 8 = 30 -k + 8 + -8 = 30 + -8 -k = 22 -k/-1 = 22/-1 k = -22

Solve this system of equations by graphing: -3x + y = 6 x + 2y = -2

Start by solving each equation for y. -3x + y = 6 -3x + y + 3x = 6 + 3x y = 3x+ 6 x + 2y = -2 x + 2y - x = -2 - x 2y = -2 - x 2y/2 = -2/2 - x/2 y = -1 - 1/2 y = 1/2x - 1 (-2, 0)

We pick two points on the line and then we subtract the y-values to get the numerator. Then subtract the x-values and make this the denominator. The formula is super handy if the points are decimals or fractions. The important thing to remember is to keep the points in the same order in the numerator and denominator.

Suppose we picked the first point to be (-4, 1), so x1 = -4 and y1 = 1. The second point is (4, 3), so x2 = 4 and y2 = 3. Plugging these into the slope formula we get: m = y₁ - y₂/x₁ - x₂ m = 1 - 3/-4 - 4 m = -2/-8 = 1/4 So m = 1/4, which is the same answer we got using the other method. Also, we totally could've switched the coordinates and said that (x1, y1) = (4, 3) and (x2, y2) = (-4, 1) instead. We can start with either point, as long as we're careful to keep both x1 and y1 on the same side of the fraction.

Terms

Terms are the separate values in an expression. Each term can be a variable, a number and a variable, or a number and many variables with or without exponents, as long as everything is being multiplied together in a single nugget of math goodness. Some of our favorite terms include: 7 7x 7x²y x⁶y¹¹z² These all count as a single term. A number next to a variable means that number and variable are being multiplied. The same is true for two variables next to each other: it means they're being multiplied. Since the × we used for multiplication in the past looks an awful lot like the variable x, we'll stop using a symbol for multiplication at all. Some call it lazy; we call it efficient.

Solve the system of equations: x + 5y = 11 2x - 3y = 4

The coefficients of x aren't additive inverses, but we can be sneaky and mold them into what we want them to be. Take the first equation and multiply each side by -2. This gives us -2(x + 5y) = -2(11), which simplifies to -2x - 10y = -22. Write this on top of the second equation: -2x - 10y = -22 2x - 3y = 4 Now the coefficients of x are additive inverses of each other, so we can work our system-solving magic. Nothing up our sleeves here... Add the equations to eliminate the x: -10y - 3y = -18 Then solve to get: y = 18/13 Now we can stick that value in for y in the first original equation to find x: x + 5 (18/13) = 11 x = 53/13 We think the solution to the system of equations is (53/13, 18/13) As always, we're going to cover our hineys and double-check this in the original equations. For the first equation, when x = 53/13, and y = 18/13, the left-hand side is (53/13) + 5 (18/13) = 53 + 90/13 = 143/13 which is 11, exactly as it should be. For the second equation, when x = 53/13, and y = 18/13, the left-hand side is 2 (53/13) - 3 (18/13) =106/13 - 54/13 = 52/13 = 4 This agrees with the right-hand side of the second equation, so we've got it. Man, being right never gets old. When we solve a system by addition, we could also say we're solving by elimination. We eliminate one variable, find the value of the other, and then find the value of the variable we eliminated. It's like the NFL playoffs. These variables are one and done.

Eliminate the variable x from the following system of equations to get an equation in terms of y. 3x + 2y = 4 2x - y = 5

The coefficients on the x are 3 and 2, which have a common multiple of 6. If we multiply the first equation by 2 and the second equation by 3, we're almost there: 2(3x + 2y) = 2(4) 3(2x - y) = 3(5) 6x + 4y = 8 6x - 3y = 15 Now the coefficients on x in the two equations are the same, which is close, but no cigar. Again, we're not referring to a literal cigar. We can't condone smoking. We want our coefficients to be additive inverses, so we can multiply either equation by -1. In other words, we need to flip all the signs in one equation. We could flip a coin to determine which one we should choose, but how do we decide which coin to flip? We'll make things easy on ourselves and go chronologically. Emphasis on "logically." If we pick the first equation, we have -6x - 4y = -8 6x - 3y = 15 Now we add the equations to find -7y = 7, and the x has been eliminated. There's one other thing that can happen: our linear system can contain fractions. When this happens, we just get rid of the fractions first and then solve the system as we've been doing. If they don't go quietly, don't be afraid to get a bit rough. Show those fractions who's boss.

y = 2x + 1

The equation y = 2x + 1 is in slope-intercept form. The coefficient of the x-term is 2, so the slope is 2. The constant is 1, so the y-intercept is 1. That means the graph passes through the point (0, 1).

Solve this system of equations for x and y: y = 1/2x + 2 -3x + y = 2

The first line is already in slope-intercept form, so we can easily plot it with a y-intercept of 2 and a slope of 1/2. The second line needs to be solved for y in order to be in slope-intercept form. We've got this. -3x + y = 2 add 3x on each side -3x + y +3x = 2 + 3x simplify y = 3x + 2 Now we can graph it with a y-intercept of 2 and a slope of 3 (or 3/1). They meet at the point (0, 2)

Solve this system of equations for x and y: y = -8 y = -4x + 4

The first line is simple to graph. It's just a horizontal line passing through the y-axis at -8. The second line has a y-intercept of 4 and a slope of -4 (or -4/1). These two lines meet at the point (3, -8), so that's our solution.

Solve this system: y = 6x - 4 y = 3x + 5

The first thing we need to do has already been done: the first equation has been solved for y. Don't you love it when someone's already come by and done the work for you? We know that y = 6x - 4, so we can substitute (6x - 4) for y in the second equation: y = 3x + 5 (6x - 4) = 3x + 5 Now we can solve the new equation for x. Start by subtracting 3x from both sides: 6x - 4 = 3x + 5 6x - 4 - 3x = 3x + 5 - 3x 3x - 4 = 5 Then add 4 to both sides and divide by 3: 3x - 4 = 5 3x - 4 + 4 = 5 + 4 3x = 9 3x/3 = 9/3 x = 3 Since a solution to a system of linear equations is a point, we need to know what y is. Until we know y, all we have is half a point, and it's difficult to win an argument with one of those. To find y, we take our value for x, stick it into either equation we like, and solve for y. If x = 3 and y = 6x - 4, then y = 6(3) - 4 y = 14 We think the point (3, 14) is the answer. To confirm this, we need to make sure this point satisfies both of the original equations. If it fails either test, we can toss it out with yesterday's garbage. Hope it likes day-old sushi. Is the point (3, 14) on the line y = 6x - 4? When x = 3 and y = 14, the right-hand side of this equation is 6(3) - 4 = 18 - 4 = 14, which agrees with the left-hand side of the equation. The point (3, 14) is on the first line. Is the point (3, 14) on the line y = 3x + 5? When x = 3 and y = 14, the right-hand side of this equation is 3(3) + 5 = 14 which agrees with the left-hand side of the equation. The point (3, 14) is on the second line. Since the point (3, 14) is indeed on both lines, it's the solution to the system of equations and the answer to all our dreams. Well, except for that one dream where our hands are giant meatballs. We still don't have an answer for that one.

How to Isolate the Variable

The most straightforward way to get a variable alone is to undo the operation that accompanies it. In the equation above, 4 is added to x. To undo this, we subtract 4 because subtraction is the opposite of addition. Here are some ways to undo other operations:

Christy is playing an online game mining precious jewels, and she comes across a mother lode of emeralds. The most she's ever mined is 4 blocks of emeralds. Write an inequality expressing how many blocks she needs to mine to beat her record.

The number of blocks Christy mines needs to be greater than 4. If x represents the number of blocks she mines, then an inequality representing this would be: x > 4

Solve for x: (2x + 6)/4 = -1

The number that's the least attached is the 4 (it's just happy not being in a relationship right now, okay?), so we first multiply each side by 4. (2x+6)/4 = -1 multiply each side by 4 [(2x + 6)/4]4 = (-1)4 simplify 2x + 6 = -4 subtract 6 from each side 2x + 6 - 6 = -4 - 6 simplify 2x = -10 divide each side by 2 2x/2 = -10/2 x = -5 and we're done Oops, we forgot to check. (2(-5) + 6)/4 = -1 (-10 +6)/4 = -1 -4/4 = -1 -1 = -1

Farmer Ted is putting up a chain link fence around a rectangular garden plot. He has 75 feet of fence and wants the garden to be 15 feet wide. Write an inequality expressing the possible length of the garden plot.

The perimeter of a rectangle is 2l + 2w = perimeter, where l is the length and w is the width. We know that the total perimeter needs to be less than or equal to 75, because that's all the fencing Farmer Ted has. So the inequality looks like this. 2l + 2w ≤ 75 We also know the width is 15, so we can swap out w for 15. 2l + 2(15) ≤ 75 Now solve this bad boy for l. 2l + 2(15) ≤ 75 2l + 30 ≤ 75 2l + 30 - 30 ≤ 75 - 30 2l ≤ 45 Divide by 2 to get l by itself. l ≤ 22.5 Since our measurements are in feet, the length needs to be less than or equal to 22.5 feet.

Translate the following verbal expression into a numeric expression: Ten less than the product of a number and 13.

The phrase "Ten less than" means 10 is being subtracted from something, so it's gotta go at the end. ___ - 10 What is 10 less than? It's less than a product, so we'll put some parentheses before the - 10. ( )( ) - 10 What is being multiplied in the product? A number (we'll call it n) and 13, so we put them in the parentheses. (13)(n) - 10 Drop the parentheses and the expression is 13n - 10, which is our final answer.

x-intercept

The point where the line crosses the x-axis. Notice that the y-value is 0.

y-intercept

The point where the line crosses the y-axis. Notice that the x-value is 0.

Notice that none of these examples have subtraction or division in them.

The properties above do NOT work with subtraction and division.

In the following phrases, we're told the operation first and then the two numbers in the operation. We just keep them in the order given.

The quotient of x and 7: x/7 The difference between x and 7: x - 7 The product of 3 and x: 3x

Slope

The slope of a line measures its steepness; the larger the absolute value of the slope, the steeper the line is. If we don't know the slope, it's usually represented by the variable m. Why? Maybe because the French word for climb is "monter," but who knows? Some mathematician probably just chose m as the lucky letter.

Graph the equation y = -½x - 2

The slope of the line is -½, which means that it slopes downhill and isn't very steep. We also know that it crosses the y-axis at -2. We'll start by plotting the y-intercept and then counting over to the next point using a slope of -½. Remember that slope equals rise over run, so this line moves downward by 1 in the vertical direction for every 2 in the horizontal. Also, since it's a negative slope, make sure that our line moves downhill from left to right.

Intercepts

The x- and y-intercepts of a line are the points where the line intercepts, or crosses, either the x-axis or the y-axis.

Solve this system of equations by graphing: y = -x + 1 x = 1/3x - 3

The x-value is 3. (3, -2)

Solve this system of equations by graphing: y = 4 y = 2x + 1

The y-value must be -1. (-1, -1)

Inequalities are exactly what they sound like: equations where the sides are "inequal" (not equal) to each other.

There are five basic inequalities that we need to be familiar with: < less than > greater than ≤ less than or equal to ≥ greater than or equal to ≠ not equal to

Divide 64abc/8abcd.

There are no variables with exponents that we need to write out, so we can go straight into reducing: Simplify that beast. 8*1*1*1/1*1*1*1d = 8/d

Multiply (7x² + 3)(7x² + 3)

There's no subtraction, so we don't need to worry about adding the opposite. Let's go with the box method again, yeah? All together: 49x⁴ + 21x² + 21x² + 9 Smash those like terms together. 49x⁴ + 42x² +9 The end.

Finally, we're getting into the kinds of problems that most people usually think of when they imagine algebra: the ones where we solve for x.

There's one extremely important rule to follow when solving all algebraic problems:

In this equation, y and x are variables and k is the constant of proportionality. Notice there's no y-intercept—they aren't allowed in proportional relationships.

They muddle things up. In this type of equation, we say that y is directly proportional to x.

What values of x will make 3/x + 1 undefined?

This is a little trickier, but not much. The question is, what would x need to be to make x + 1 = 0? If we remember our integers, we know that -1 + 1 = 0. If x = -1, then we get a denominator of 0 and an undefined expression. Check it: 3/(-1) + 1 = 3/0 Basically, all we're doing is setting the denominator equal to 0 and then solving for x. Here's the trick, though. We've solved for the number we don't want x to be, ever. Every other time we solve for x we're solving for the number x is, not the number it can't be. But when we're trying to find where the expression is undefined, we're trying to find what we don't want x to be (unless we're trying to make a black hole). Though we will say, it's pretty funky what getting close to 0 in a denominator does to the graphs of equations. But that's for another day.

Horizontal lines are all in y = A form, where A is any real number.

This is because we're graphing all points where y equals some number.

Multiplying Binomials

This is the last type of multiplication that we're going to look at in this unit. The good news is that there's nothing new to learn here. All we're really doing is applying the distributive property twice. The most important part of multiplying two binomials is to make sure that we multiply each term in the first factor by each term in the second. There are several operations that need to happen, and there are different ways to track each operation to make sure we get 'em all. We're going to look at three methods. They all do the same math and get the same answer, but in different orders, so we can pick one and stick with it.

Solve for z: 100/z = -4

This one is different from the first division example; here the variable is hanging out in the denominator. To solve for z, we need to get the variable back on top. 100/z = -4 (multiply each side by z) (100/z)z = -4(z) (the z's on the left cancel out) 100 = -4z (divide each side by -4) 100/-4 = -4z/-4 (the -4's cancel each other) -25 = z Now, don't forget to check: 100/-25 = -4 -4 = -4

Solve for y: 7 -y = 10

This one's a bit trickier. We're going to show you how to do this two different ways. 7 - y = 10 (add -7 to each side) 7 - y + (-7) = 10 + (-7) (the negative of y is 3) -y = 3 (divide both sides by -1) y = -3 Here's the other way: 7 - y = 10 (add y to each side) 7 - y + y = 10 + y (add -10 to each side) 7 + (-10) = 10 + y + (-10) -3 = y Now check the answer: 7- (-3) =10 7+3 = 10 10 = 10

Solve this system of equation using addition: y = 3x -y + 4 = x

This time, we're gonna add the equations: lump all the stuff on the left of the equal signs together, and ditto for the right side of the equal signs. y + -y + 4 = 3x + x 4 = 4x 1= x Plug 1 in for x in the first equation, and we get y = 3(1) = 3. So the solution is (1, 3).

0 and Undefined Slopes

To find the slope of a vertical line, we can start with either point, but let's pick (4, -1) as the first point and (4, 3) as the second point. Plug these into the slope equation. m = -1 - 3/4 - 4 = -4/0 Wait, something's wrong here—zeros aren't allowed in a denominator. What is the steepness of a vertical line? How can we measure the steepness of an infinitely steep line? We can't. To make our math brains feel better, we call this steepness undefined. All vertical lines have undefined slopes.

Although we really only need two points to make a line, finding a third one is often a good idea. If all three points lie in a straight line, we can feel confident that we didn't make a mistake. If the third point doesn't fit our line, we check our work and try again. Let's start with a simple example: y = 2x + 1

To find three points on this line, we pick any values we want for one variable, plug them into the equation, then solve for the other variable. Since y is already isolated in this equation, it's a good idea to start by picking values for x. This will give us a value for x and one for y, which we can plot as an (x, y) point! Here's a tip: in the beginning, go easy on yourself and pick nice and simple values for x, like -1, 0, and 1.

Solve for x: x + (-8) = -20

To isolate x, we can do one of two things: subtract -8 or add +8. We prefer adding the opposite. x + (-8) = -20 (add 8 to each side) x + (-8) + 8 = -20 + 8 (simplify) x = -12 To check, let's plug -12 in for x. (-12) + (-8) = -20 -20 = -20 It works!

Solving Systems of Linear Equations by Substitution

To solve linear systems by substitution, we solve one equation for one variable and then use that information to solve the other equation for the other variable. It's exactly the same as when a basketball team makes a substitution, except with less basketball and more math. Let's do a couple of examples and see what happens.

Some words can be translated into math, and some math can be translated into words, and neither require the babel fish from The Hitchhiker's Guide to the Galaxy. When we translate words to math, most are replaced by symbols in the order that they're written.

To translate "seven plus a number," we replace the word seven with the number 7, replace the word plus with the addition sign, and replace the phrase "a number" with a variable like x to get: 7 + x

Write this equation in standard form. y = 3x + 7

To write this equation in standard form, we move the x-term to the left side of the equal sign so that both the x-and y-terms are together. Trust us, they're happier that way. Subtract 3x from both sides. y - 3x = 3x + 7 - 3x y - 3x = 7 Beautiful. One standard-form equation, served up hot with fries on the side.

Graph the equation 4x + y = 7

Unfortunately, this equation is not given to us in slope-intercept form ☹. However, we can solve the equation for y and manipulate it into this form. 4x + y = 7 subtract 4x from each side 4x +y - 4x= 7 - 4x simplify y = 7 - 4x switch the 7 and 4x but keep the appropriate sign y = -4x + 7 Now we can see that the slope of this equation is -4, also known as -4/1 when written as a fraction, and the y-intercept is 7. We'll start by plotting the y-intercept of 7 and then counting over to the next point using a slope of -4/1.

Solve the system of linear equations: y = 3x + 2 x - y = 4

We add the equations to find that y + x - y = 3x + 2 + 4. This simplifies to x = 3x + 6. All of a sudden the variable y is gone. We don't know where it went, but hopefully it's in a better place. We now have an equation we can solve for x. Doing this yields x = -3. Now we can put -3 in for x in the first equation to find y: y = 3(-3) + 2 = -7 The solution to the system appears to be (-3, -7). Let's check this answer in the original equations. We want to check it before we wreck it. For the first equation, when x = -3 and y = -7, the left-hand side is -7 and the right-hand side is 3(-3) + 2, which also happens to be -7. These values work in the first equation. For the second equation, when x = -3 and y = -7, the left-hand side of the equation is x - y = (-3) - (-7), which is 4, the same as the right-hand side of the equation. These values work in both the equations, so the solution really is (-3, -7). The previous example worked because the the coefficients of y in the two different equations were additive inverses of each other. The first equation had a positive y, and the second equation had a negative y. y = 3x + 2 x - y = 4 When we added the equations, we eliminated the variable y. Now y sleeps with the fishes. This example was one of the kindest ones we could have given you, because the coefficients of y in the two equations were already additive inverses. In general, we need to do much more legwork to get coefficients that are additive inverses. Once we're done with our legwork, we can focus on our glutes.

We can find the solution to a system of linear equations by graphing like a boss, but there are other ways, too. We'll get into these other methods in more detail when we're past pre-algebra, but here's a sneak peek.

We can also solve systems of linear equations by substitution and by addition. And thank goodness, because graphing a bunch of equations with fractions is no fun.

How to Graph Inequalities

We can graph inequalities on a number line to get a better idea of how they're behaving. Just follow these steps. 1. Find the number on the other side of the inequality sign from the variable (like the 4 in x > 4). 2. Sketch a number line and draw an open circle around that number. 3. Fill in the circle if and only if the variable can also equal that number. 4. Shade all numbers the variable can be.

Alex sells books online. She makes a flat profit of $2.00 per book, but she needs to pay $4.00 per day to Paypal for using the app on her website. How many books does she need to sell to make at least $120.00 per day?

We can multiply the number of books sold by the profit per book and then subtract $4.00 to figure out how much she makes in a day. Let's say b is the number of books Alex sells per day. 2b - 4 = profit each day Since we want the profit to be at least $120.00, we want it to be greater than or equal to than $120. We can rewrite this as an inequality. 2b - 4 ≥ 120 We solve for b by adding 4 to both sides of the inequality and then dividing by 2. 2b - 4 + 4 ≥ 120 + 4 2b ≥ 124 b ≥ 62 Alex needs to sell at least 62 books per day to make $120.00 , but hopefully she'll sell even more.

Which expression is undefined when x = -5? a. x + 5/4 b. x + 5 c. 4/x + 5 d. 4/x - 5

We can replace the x in each expression with -5. If it makes the denominator equal 0, then the expression is undefined at x = -5. Let's take it one equation at a time. a. x + 5/4 = -5 + 5/4 = 0/4 = 0 Nope, not undefined. The fraction equals 0, but that's not the same thing as dividing by 0. b. x/5 = -5/5 = -1 Nope, still good. c. 4/x + 5 = 4/-5 + 5 = 4/0 = undefined Ah, there we go. A 0 in the denominator means our fraction implodes in a puff of undefined smoke. And just for fun... d. 4/x - 5 = 4/-5 - 5 = 4/-10 That one's totally defined, too. The only answer is c.

Standard Form

We can write an equation in standard form by moving the x and y terms to the same side of the equal sign. Standard form looks like this, where A, B, and C are real numbers. Ax + By = C The standard form of y = x + 3, for example, is y - x = 3.

Audrey II is a man-eating plant living in the Little Shop of Horrors and it needs blood, and lots of it. Luckily its caretaker, Seymour, has discovered a blood substitute to feed Audrey II. Now that death is out of the way, he can figure out how fast Audrey II grows. Audrey II is 10 inches tall and it grows a steady 3 inches a day for x days. How tall will it be in 7 days? How tall in 14 days?

We can write an expression for the growth rate of Audrey II by adding its current height (10 in.) to the product of how much it grows each day (3 in.) and the number of days it grows (x). 10 + 3x To be consistent with traditional algebraic order, we'll write the variable term (3x) before the constant term (10). 3x + 10 Now that we have our expression, we can use it to see how tall Audrey II will be in 7 days and how tall it will be in 14 days. When x = 7 we have: 3(7) + 10 = 21 + 10 = 31 So in 7 days, Audrey II will be 31 inches tall. Now try it with x = 14. 3(14) + 10 = 42 + 10 = 52 In 14 days, Audrey II will be 52 inches tall. Yikes!

Tickets to a concert cost $25.00 each, including tax. Xavier has $150.00 and wants to get tickets for himself and some friends. How many friends can he invite?

We first write this as an equation and then solve. The total cost is $150, which equals the product of the number of tickets, t, and the cost per ticket, $25. 150 = 25t Solving for t, we get 150 divided by 25. 150 = 25t 150/25 = 25t/25 150/25 = t 6 = t Xavier can buy 6 tickets total (including one for himself), so he can invite 5 friends.

Example 1: One-Step Inequalities A bag of candy is split between us and our little brother. The bag says it has at most 28 pieces of candy in it. There are 15 candies in our bag, and x candies in our brother's bag, and we have to make sure he doesn't have more than us (of course). We can write an inequality expressing the number of candies in the bags.

We know that our bag has 15 candies. We know that his bag has x candies. We know that up to 28 candies were split between both bags, no more. Since we know that x + 15 can't be more than 28, it must be less than or equal to 28. x + 15 ≤ 28 Now that we have an inequality expressing our candy, we solve for x. To do that, we subtract 15 from both sides. x + 15 ≤ 28 x + 15 - 15 ≤ 28 - 15 x ≤ 13 Sweet, literally. Our brother has 13 or fewer pieces of candy, so no way does he have more candy than us. Until, in a moment of weakness (er, kindness) we share a few of our pieces with him. Now it's even sweeter.

Example 2: Two-Step Inequalities A candy store owner saw us share our candy with our brother and was so impressed he gave us a $30.00 gift card. Tax free even! We decide to buy a giant candy bar for $13.00 and then some lollipops with the remaining money. If each lollipop is $0.90, we can write and solve an inequality expressing how many lollipops we can buy.

We know that the amount we spend needs to be less than or equal to $30.00. We know we want to buy x lollipops. We know that each lollipop is $0.90, so the total cost of the lollipops is $0.90x. We know the candy bar costs $13.00. So the total cost of the candy is $13 + $0.90x, and this needs to be less than or equal to $30.00. 13 + 0.90x ≤ 30 Now we solve for x, which is the number of lollipops we can buy. We want to get x by itself, so start by subtracting 13 from both sides. 13 + 0.90x ≤ 30 13 + 0.90x - 13 ≤ 30 - 13 0.9x ≤ 17 Divide both sides by 0.9 to finish up. We recommend nabbing a calculator for this part. 0.9x /0.9 ≤ 17/0.9 x ≤ 17/0.9 x ≤ 18.89 We need to buy less than 18.89 lollipops, so we can buy 18 whole lollipops and have some change to spare.

Solve for x: (x+5)/2 = 10

We should get rid of the 2 first, since it's not as attached to the variable as the 5. (x+5)/2 = 10 multiply each side by 2 [(x+5)/2]2 = (10)2 the 2's on the left cancel x + 5 = 20 subtract 5 from each side x+5 - 5 = 20-5 and simplify x=15 And check: (15 + 5) /2 =10 20/2 = 10 10 = 10

Solve for y: 14x - 3y + m = -12m - 3x - 4y

We start by moving all of the terms without a y to the right of the equal sign. 14x - 3y + m = -12m - 3x - 4y subtract 14x from both sides 14x - 3y + m -14x = -12m = 3x - 4y -14x simplify -3y + m = -12m -3x - 4y - 14x combine like terms -3y + m = -12m - 17x - 4y subtract m from both sides -3y + m - m = -12m - 17x - 4y - m combine like terms -3y = -13m - 17x - 4y add 4y to both sides -3y + 4y = -13m - 17x - 4y + 4y simplify y = -13m - 17x y is now all alone on the left of the equal sign—it's lonely at the top (or on the left)—and we've solved for y.

Multiply 15(2x + 3y - 1)

We still start by changing the subtraction symbol to adding a negative. 15(2x + 3y + -1) We need to distribute the 15 to each term inside the parentheses. 15(2x + 3y + -1) 15(2x) + 15(3y) + 15(-1) Finish up with some good old-fashioned multiplication and addition. 15(2x) + 15(3y) + 15(-1) = 30x + 45y - 15

Solve for k: (2k +8)/k = 4

We threw this one in here to keep you on your toes. At first glance it looks like all the variables are on the left side. However, when you multiply each side by k (to get the variable out of the denominator), there will be variables on each side. (2k +8)/k = 4 multiply each side by k [(2k +8)/k]k = (4)k the k's on the left cancel 2k + 8 = 4k subtract 2k from each side 2k + 8 - 2k = 4k - 2k simplify 8 = 2k divide each side by 2 8/2 = 2k/2 4 = k Check! (2(4) + 8)/4 = 4 (8+8)/4 = 4 16/4 = 4 4 = 4

Graph the following inequality: y < -2 and y > 1

We're dealing with an "and" inequality, so we only graph the numbers that overlap. Hmm...there don't seem to be any numbers that overlap. This means that no numbers satisfy both conditions. No Solution!

Create a graph and write an equation for the linear relationship represented by the following points: (2, 5), (3, 6), (4, 7), (5, 8).

We're gonna need the slope, so let's start by tossing those first two points into our slope formula blender and making a delicious slope smoothie. m = y₁ - y₂/x₁ - x₂ m = 5 - 6/2 - 3 = -1/-1 = 1 Now we can graph the line and find the y-intercept. The y-intercept is 3 and the slope is 1, so the equation of the line is y = x + 3.

Method 2: Distributing the second factor to both terms in the first factor

We're starting with the exact same problem. (x - 3)(2x + 1) = (x + -3)(2x + 1) In this method, we first distribute x to the entire binomial (2x + 1), keeping the addition sign between them. Then we distribute the -3 to the entire binomial (2x + 1). (x)(2x + 1) + (-3)(2x + 1) Now we run through the distributive property a second time: distribute the x to both 2x and 1, and distribute -3 to both 2x and 1. (x)(2x) + (x)(1) + (-3)(2x) + (-3)(1) = 2x² + x + (-6x) + (-3) = 2x² + x - 6x - 3 And once again, we combing those x terms to get our final answer. 2x² + x - 6x - 3 = 2x² - 5x - 3 Nice! We got the same answer as we did using the first method. But wait, there's more. We've got another trick up our sleeves.

This multi-step business may be a bit more complicated than what we've already been doing, but it's nothing we can't handle. It just involves three or more of the same kinds of steps.

We've already seen how to solve for x in an equation that has two operations, like addition and multiplication. Doing PEMDAS backwards, we usually get rid of the numbers being added and subtracted first, and then the numbers being multiplied and divided. The key word here is usually because this ain't no hard-and-fast math rule.

A Summary, So Far

We've now used substitution to successfully find the point of intersection for two lines that intersect exactly once. Let's tidy things up a bit and figure out the general steps we need to take for this sort of problem. Once we're done, we should also tidy up the living room. It's great that you wanted to build a fort out of the couch cushions, but people have to live here. 1. Solve one equation for one variable. 2. In the other equation, perform substitution to get rid of the variable we solved for in step 1. 3. After substituting, solve the other equation. 4. Find the value of the variable we solved for in step 1. 5. Check that the answer works in both original equations.

Linear Equations

We've now worked pretty extensively with equations containing two variables. All the equations we've looked at so far have represented straight lines, so we call them linear equations. There are several different forms of linear equations, but they all have one thing in common: the variables don't have exponents (other than the invisible 1). Ready for a newsflash? If any of the variables do have exponents, the equation isn't a straight line. It's a curve. Wild, right?

Plot the points (-4, 4), (1, 4), and (1, -1). What are the coordinates of an additional point that would make a square when all four points are connected? a. (-4, -1) b. (-4, -4) c. (-1, 4) d. (-1, -4) e. (-1, -1)

a. (-4, -1)

Solve the equation 2c - 10/4 = c for c. a. -5 b. 5 c. 0 d. All real numbers e. No solution

a. -5

Simplify the expression 3(2x + 6) - 4(x - 7). a. 2x + 46 b. 2x - 10 c. 6x - 14 d. 10x + 46 e. 10x - 10

a. 2x + 46

Solve the inequality y > -1 or -2y < -4. a. y > -1 b. y > -2 c. y < -1 d. y < 2 e. y > 2

a. y > -1

Solve this system of equations. y = 2/3x - 2/3 y = 1/2x a. (0, -4) b. (4, 2) c. (0, 0) d. (4, 0) e. No solution

b. (4, 2)

Divide the expression 24y³ - 12y² + 8y/-4y a. 6y² - 3y + 2 b. -6y² + 3y - 2 c. -6y² - 3y - 2 d. 6y² + 3y + 2 e. Cannot be simplified.

b. -6y² + 3y - 2

The point (2, -3) is translated five spaces to the left and four spaces up. What quadrant is it now in? a. Quadrant I b. Quadrant II c. Quadrant III d. Quadrant IV e. At the origin

b. Quadrant II

Solve the inequality 3z + 2 - (2z + 4) ≥ -z for z. a. z ≥ 0 b. z ≥ 1 c. z ≤ 1 d. z ≥ -1 e. z ≤ -1

b. z ≥ 1

Divide the expression 180a⁴b⁵c/-20abc² a. a³b⁴/-9c b. -9(abc) c. - 9a³b⁴/c d. -9c/a³b⁴ e. -9ac/b⁴

c. - 9a³b⁴/c

The perimeter of a rectangular pool is 60 feet. The length is twice as long as the width. Find the length. a. 10 ft b. 15 ft c. 20 ft d. 25 ft e. 30 ft

c. 20 ft

Solve the equation -6(2x + 8) = -12x + 17 for x. a. 0 b. -1 c. 1 d. All real numbers e. No solution

e. No solution

Solve this system of equations. y = x x - y = -2 a. (1, 1) b. (0, 0) c. (0, 1) d. (2, 2) e. No solution

e. No solution

Solve the inequality -2(x - 10) ≤ 8 for x. a. x = 6 b. x ≤ -6 c. x ≥ -6 d. x ≤ 6 e. x ≥ 6

e. x ≥ 6

For example, let's look at -2x + 3 > 5

solve like you would -2x + 3 = 5 -2x + 3 > 5 subtract 3 from each side -2x + 3 - 3 > 5 - 3 simplify -2x > 2 divide each side by -2 -2x/2 > 2/-2 x > -1 switch the sign from > to < x < -1

But since we're blasting ahead in math...

we'll soon be graphing on charts that look like this:

Graph the line y = 4x - 3

x = 0 y = 4(0) - 3 y = 0 - 3 y = -3 (0, -3) x = 1 y= 4(1) - 3 y = 4 - 3 y = 1 (1, 1) x = 2 y = 4(2) - 3 y = 8 - 3 y = 5 (2, 5) Plot and connect. Did you get a straight line?

Simplify: (2x² + 5x - 1) + (5x² - 2x + 9)

(2x² + 5x - 1) + (5x² - 2x + 9) 7x² + 3x + 8

Simplify: (7a +3b - 2)n- (4a - 7b - 5) -(a + b + 1)

(7a +3b - 2)n- (4a - 7b - 5) -(a + b + 1) 2a + 9b + 2

Use the distributive property to simplify -3(2z - 4)

-3(2z) - -3(-4) -6z + 12

Monomial Expression

1 Term Example: xy

Polynomial

A polynomial is an expression that's made up of constants and/or variables. All the expressions we've been dealing with so far have been polynomials: 5x + 17 and 18xy2 - 17xy + 19y are both polynomials, for example. And we saw from our handy chart earlier that a monomial is an expression that's made of a single term, like 5x. ("Mono-" just means "one.") When we learned about the distributive property, we were multiplying polynomials, but now we'll look at this a bit deeper. Look at the examples carefully and make note of the exponents. Remember: 5xy means 5 times x times y. Again, it's helpful to think of subtraction as adding a negative: (x - 5) is the same as x + (-5). This will help us keep track of which terms are negative and which are positive.

Associative Property

Addition: x + (y + z) = (x + y) + z Multiplication: x(yz) = (xy)z The Associative Property allows us to move the parentheses to a different pair of numbers as long as everything is being multiplied or everything is being added. For example: 7 + (x + 10) is the same thing as (7 + x) + 10.

Commutative Property

Addition: x + y = y + x Multiplication: xy = yx The Commutative Property states that we can add or multiply numbers in any order. For example: 7xy is equal to y7x and yx7. In Algebra we almost always put the coefficient in front of the variables, but just for consistency, not because it needs to be that way mathematically.

Evaluate the following expression when x = 2, y = -3, and z = -1. 2x + 3y - z

All we need to do is replace each variable with the assigned number. 2(2) + 3(-3) - (-1) Now simplify that junk. 4 + (-9) + 1 Last but not least, add everything up. -5 + 1 = -4

Look Out!

An exponential pattern is actually a type of geometric pattern. However, to help explain things, we made them a subcategory.

Let's get down to business.

An expression is made up of terms.

Find the missing number in the pattern: 3, 9, 81, ___, 43,046,721.

As mentioned before, there are three basic types of patterns: arithmetic, geometric, and geometric-exponential. Let's try all three and see if one fits. Here we can see that geometric-exponential patterns are also geometric. We can use either of these patterns to fill in the blank, which gives us our missing number: 6561.

Things can seem a little more complicated when dealing with subtraction. We just need to be extremely careful to keep the operations with the correct terms.

Check it out: 2y + 5x - 8y In this example, there are two terms that can be combined (2y and 8y). However, it's "minus 8y" and we reeeally must be careful to keep the subtraction sign: 2y - 8y = -6y. This expression simplifies to: -6y + 5x It's also worth noting that the order of addition doesn't matter. This expression could also be written as 5x + (-6y) or 5x - 6y.

6.4

Combining Like Terms

Combining Like Terms

Combining like terms is pretty chill, as long as we're careful with our negative and positive numbers. (For a quick review, check out adding integers and subtracting integers.) When adding and subtracting like terms, all we really need to do is combine the coefficients. For example, we can simplify the expression 3x + (-9x) by combining both of those x terms. 3x + -9x = -6x Both terms have an x with no exponent, so we add their coefficients to get -6x.

6.6

Distributive Property

6.3

Evaluating Algebraic Expressions

Geometric

Example: 1, 2, 4, 8, 16... Solution: Multiply the previous number by 2. Example: 1000, 100, 10, 1... Solution: Divide the previous number by 10.

Arithmetic

Example: 1, 3, 5, 7, 9... Solution: Add 2 each time. Example: 99, 90, 81, 72... Solution: Subtract 9 each time.

Geometric - Exponential

Example: 2, 4, 16, 256... Solution: Square the previous number. Example: 1, 4, 9, 16, 25... Solution: 1², 2², 3², 4², 5²...

Occasionally terms, like the last one above, seem to be missing a coefficient. Here's the thing: they actually do have a coefficient of 1. It's just too lazy to show up. Since 1 times anything is just that anything, we usually don't write 1 as a coefficient, but it's always there.

Example: x is the same thing as 1x, and x³y²m is the same thing as 1x³y²m.

Multiply x(2x)(3x)(4x)

First multiply all of the coefficients: 1 x 2 x 3 x 4 = 24 Next we need to multiply the variables: xxxx = x⁴ Put it all together, and our answer is 24x⁴

If there's more than one term separated by plus or minus signs, then we have an expression.

For example: 5x²y - 3xy + y + 5 This expression has four terms: 5x²y, -3xy, y (or 1y), and 5.

Algebraic terms can, and often should, be combined and simplified.

However, only terms that are "like," meaning that they have the exact same variables and hairdo, can be added or subtracted. Furthermore, the variables need to have the same exponent to be "like": xy² and xy are not like terms, since y is squared in the first term.

Simplify the expression: 4x + 3y + x - 7y - 3

In this expression, there are five terms. Two have an x variable, two have a y variable, and one is a constant. Let's take a peek at those x-terms first. 4x + 3y + x - 7y - 3 We have 4 x's in the first term, and one more x in the third term. Yeah, the x by itself is missing its coefficient 1, but it's with us in spirit. So we can combine the x's to get 4x + x = 5x. We also have 3 y's and -7 y's. 4x + 3y + x - 7y - 3 Combined, they make 3y - 7y = -4y. Since there's only one constant, -3 doesn't get to combine with anything else. Constants often fly solo. With everything combined, we've got the simplified expression 5x - 4y - 3.

Simplify 4xy + 5x - 13y + 10xy - y

In this expression, there are two like terms with the variables xy. 4xy + 5x - 13y + 10xy - y Combined, they make 4xy + 10xy = 14xy. The terms with just an x and just a y are not the same as the terms with an x and a y together. All the variables need to match, or we don't have like terms. There are two terms with the variable y, and both are negative. 4xy + 5x - 13y + 10xy - y Smash them together to get -13y - 1y = -14y. There's only one term with an x, so it doesn't combine with anything. Add 'em all up and we get 14xy - 14y + 5x.

Find the additive inverse of -4 and prove that it's the inverse by adding to get the identity.

It's Opposite Day up in here. The additive inverse of a number is the same number with the opposite sign, so the additive inverse of -4 is 4. If we add a number and its inverse, we should get 0, the additive identity. Since -4 + 4 = 0, we know that 4 is in fact the additive inverse of -4.

Which of the following values make the equation x4 - 1 = 0 true? x = -2, -1, 0, 1, 2

Just like the previous problem, it's probably useful to make a chart. x = -2 (-2)⁴ - 1 16 - 1 15 x = -1 (-1)⁴ - 1 1 - 1 0 x = 0 (0)⁴ = 1 0 - 1 -1 x = 1 (1)⁴ - 1 1 - 1 0 x = 2 (2)⁴ - 1 16 - 1 15 Now we can see that when we plug in both -1 and 1, the value is 0, making the equation true.

Variables

Letters that represent an unknown or changing number.

Like terms

Like terms are terms that have the same variables, including the exponents that go with those variables. The variables can be in different orders and have different coefficients, but they all need to be there. Examples: 3xy and -5xy are like terms (same variables). 3xy and -5xym are not like terms (the second term has a variable that the first doesn't). -2m²xh and 4m²hx are like terms (same variables and exponents, just in different order). -2m³xh and 4m²xh are not like terms (the variable m has different exponents in each term).

6.7

Multiplying Monomials

In these translations, we'll use the letter x to represent the variable, though any letter, symbol, or emoticon would work.

Smiley faces and hearts, anyone?

Find the next two numbers in this pattern: 1, 2, 8, 48, 384...

Some of you may have figured out the pattern by just looking at it, the rest of us may need a little help. Let's start by writing these numbers in a chart and looking at their differences. Hmm... addition doesn't seem to be working, so let's try multiplication. Seeing the pattern yet? Each time we multiply by the next even number. Here is how it would continue: The next two numbers in the pattern are 3840 and 46,080.

Fill in the blanks to illustrate the associative property of multiplication. (3x)y = 3( )

The associative property of multiplication says we can shift parentheses around different numbers as long as everything is being multiplied. Moving the parentheses from the 3x to the xy shows the associative property. (3x)y = 3(xy)

Find the multiplicative inverse of 4/5 and prove that it's the inverse by multiplying to get the identity.

The multiplicative inverse of any number is its reciprocal. In other words, flip the numerator and denominator but keep the sign the same. So the multiplicative inverse of 4/5 is 5/4. Multiply 'em together. (4/5)(5/4) = 20/20 = 1 Nice—since 1 is the multiplicative identity, we know these numbers are inverses.

Simplify: (2x² + 3x) + (-6x - 5x²)

The parentheses in this expression are not necessary, since it doesn't change how we treat each expression. We can rewrite it without 'em, then mark each set of like terms. The tricky part with this problem is to keep the correct addition and subtraction signs with each term. Combine everything and we get our final answer. -3x² + -3x

A triangle has no diagonals, while a quadrilateral has 2, a pentagon has 5, and a hexagon has 9. Without drawing the figure, how many diagonals will a septagon (7-sided figure) have?

The pattern is +2, +3 +4, +5. A septagon has 14 diagonals.

When multiplying two of the same variables, add the exponents. Remember that the exponent on x is an invisible 1. x * x² = x¹ * x² = x³ x * xx = xxx = x³

The reason for this is that x² is really just x times x, and x times x² is x times x times x, or xxx, which equals x³ (since there are three x's). The exponent tells us how many variables to multiply together.

Distribute 4x(2y + 5)

The terms inside the parentheses are being added, not multiplied or divided, so we can distribute the 4x to each of them. Distribute! Distribute like the wind! 4x(2y) + 4x(5) Multiplying the coefficients and then the variables, we get our final answer. 8xy + 20x

Simplify: 3xy + 3x²y + -6xy +7xy²

There are only two terms in this expression that are alike: 3xy and -6xy. 3xy + -6xy is -3xy, so our simplified expression will be: -3xy + 2x²y + 7xy²

Simplify: (4ab + 5b) - (2ab + 3a - b)

There are two things to be careful with in this problem. First, each term in the second expression is being subtracted, so we've gotta "distribute" the subtraction sign to each term when removing the parentheses. 4ab + 5b - 2ab - 3a + 1b Notice also that when we distribute the negative sign to another negative sign, it becomes addition. Now we'll mark our like terms. The second thing we need to be cautious with is the last term, b. Even though it doesn't show a coefficient, it's the same as 1b. Our simplified expression is: 2ab + 6b - 3a

Expressions are made of variables, or letters that take the place of unknown numbers. But what if we know the numbers? We can take out the variable, replace it with the number, and do the math.

This is called evaluating an algebraic expression.

Multiplying a Monomial by a Polynomial

This is the same thing as the distributive property that we just learned. Let's say we want to multiply 4x(6 - 2y). First we're going to change the subtraction symbol to adding a negative. 4x(6 + -2y) Next we distribute the 4x. 4x(6 + -2y) = 4x(6) + 4x(-2y) 24x + -8xy Rewrite it again without the whole adding-a-negative thing to get our final answer: 24x - 8xy.

Find the values of the following expression for x = -2, -1, 0, 1, and 2. 3x² + 1

This question is asking us to evaluate the expression for five different values x = -2 3(-2)² + 1 3(4) + 1 12 + 1 13 x = -1 3(-1)² + 1 3(1) + 1 3 + 1 4 x = 0 3(0)² + 1 3(0) + 1 0 + 1 1 x = 1 3(1)² + 1 3(1) + 1 3 + 1 4 x = 2 3(2)² +1 3(4) +1 12 + 1 13 The answers can be expressed as ordered pairs, with the x-value first and the answer second. Here are the answers in this form: (-2, 13), (-1, 4), (0, 1), (1, 4), and (2, 13)

Which property is illustrated below? xy + ac = ac + xy

We first need to figure out how the expression on the left is different from the expression on the right. Notice the terms have switched places like a couple of mischievous toddlers. This illustrates the commutative property of addition.

-z(-18 - z)

We now distribute -z to both -18 and -z, or multiply it by both of them. We now distribute -z to both -18 and -z, or multiply it by both of them. -z(-18) + -z(-z) Those double negatives turn into a positive, leaving us with 18z + z².

If we have the expression 3x + 1 and we find out that x = 10, we can evaluate it. The expression means "three times whatever x is, plus one."

We rewrite the expression, replacing the x with 10 in parentheses, and do the math. 3x + 1 = 3(10) + 1 = 30 + 1 = 31

Evaluate 4xy + 3x if x = 8 and y = 5.

We take out all the variables and put in the numbers in parentheses. Replace x with 8 and swap out y with 5. 4xy + 3x = 4(8)(5) + 3(8) = 160 + 24 = 184 Don't Forget: we'll need our faithful friend PEMDAS (a.k.a. the Order of Operations) to make sure we do the math in the right order.

Good news !!!

You've actually been working with algebra since you were three and began to notice patterns (red dog, blue cat, red dog, blue cat...). The patterns we're going to work with now are just a little more complex and may take more brain power. Patterns are the beginning of algebra. There are endless types of patterns and methods for solving patterns. We've mostly been working with simple arithmetic (patterns involving adding or subtracting a number each time) or geometric patterns (ones involving multiplying or dividing by a number). Here are three common types of patterns we've seen.

Common Words and Phrases for: Subtract

difference subtract less than take away

Common Words and Phrases for: Divide

divided by quotient split share distribute

eighty less than a number

x - 80

A number is negative ten.

x = -10

A number is equal to the sum of twice the number and negative three.

x = 2x + (-3)

This one is insanely important when working with algebraic expressions. The distributive property basically says this:

x(y+z) = xy +xz and x(y-z) = xy - xz

However, the distributive property does not work when the variables inside the parentheses are being multiplied or divided.

x(yz) ≠ xy(xz) and x(y/z) ≠ xy/xz

Evaluate the expression x³ + x² + x + 1 for x-values equal to -2, -1, 0, 1, and 2.

x³ + x² + x + 1 x = -2 -2³ + -2² + -2 + 1 -8 + 4 + -2 + 1 -4 + -2 + 1 -6 + 1 -5 x = -1 -1³ + -1² + -1 + 1 -1 + 1 + -1 + 1 0 + -1 + 1 -1 + 1 0 x = 0 0³ + 0² + 0 + 1 0 + 0 + 0 + 1 1 x = 1 1³ + 1² + 1 + 1 1 + 1 + 1 + 1 4 x = 2 2³ + 2² + 2 + 1 8 + 4 + 2 + 1 12 + 2 + 1 14 + 1 15 (-2, -5), (-1, 0), (0, 1), (1, 4), and (2, 15)

The formula for the surface area of a cone is πrs + πr², where r is the radius of the base and s is the length of the slant. Find the surface area of a cone with radius 10 cm and a slant of 12 cm (use 3.14 for π).

πrs + πr² (3.14)(10)(12) + (3.14)(10²) (31.4)(12) + (3.14)(100) 376.8 + 314 690.8 cm²