statistic mod 3 ;part 1
A solid line in the graph of an inequality indicates that the value of y can be equal to the values on the line. True or false
. True Correct. A solid line indicates that the inequality used was ≤ or ≥ . In both cases, the y values on the line are included.
Combining Like Terms to Solve Equations
Solve for x . 4x+8−3x=2 In the above equation, there is a set of like terms: 4x and −3x . Just like in an expression, we can also combine these in an equation to solve for x.
To write x≤2
use an open parenthesis with −∞ and a closed bracket with 2 : (−∞,2] .
combining like terms
"Like terms" in an expression should be combined, whenever possible, in order to simplify the expressions - a process referred to as combining like terms. Simplifying an expression by combining like terms* can be helpful for a number of reasons, including solving equations. To understand how we can combine these terms*, or why we even want to combine these terms, let's assign meaning to this variable. Let's say the variable j represents jars. Replacing j with jars, the expression reads as follows: (jar)+2(jar)
What is the constant in the expression below? 5/x2+15x−3.4
-3.4 Correct. The answer is d. The constant is the value at the end of an algebraic expression. It is not part of a product with any variables.
Simplify the inequality: (Enter your answer with the x variable on the left) −3x−8>x
. Bring the x over to the right side to get −8>x+3x Simplify to get −8>4x Divide both sides by 4 −8÷4>4x÷4 −2>x Lastly, switch sides so that x is on the left, remember to switch the greater-than sign to a lesser-than sign. x<−2
Which of the equations below is a linear equation? a. 4g5=8v−8w b. 3g−7x=12 c. 4w2x=3 d. g2=21
3g−7x=12 Correct. The answer is b. The g and x both have a degree of one, so this is a linear equation.
What is the slope of the following linear equation: −5x+y+8=0
5
Simplify the inequality: (Enter your answer in slope-intercept form) −y+3x>−2y−7
Bring the −2y over to the left side and the +3x over to the right side to get y>−3x−7
If the rise =5 and the run =1 , what is the slope of the line?
Correct. 5. If the rise =5 and the run =1, then 51=5. The slope is 5.
In the following inequality, the sign would need to be flipped when isolating x on the left side. True or false? 6x−1/2<x+1
False Correct. This statement is false. After multiplying both sides by 2, the inequality reads: 6x−1<2x+2 Subtracting the 2x from the 6x leaves a positive coefficient on the left side and that leads to dividing by a positive number. The sign would not be flipped
Inequalities with One Variable Example 1
For an example, an inequality reads x+6>2 . To solve for x , manipulate the inequality just like you would a regular equation. bove, we've solved the inequality; x>−4 . Though we do not know the exact numerical value of x , we do know that any value it does hold, in this case, is greater than −4 .
The division principle of equality*
If both sides of an equation are divided by the same non-zero number, the result is an equivalent equation.
y=4x+10
This first example is written already in slope-intercept form; the slope is equal to 4 and the y -intercept is equal to 10 .
True or False? All constants are rational numbers.
This is a false statement. A constant can be rational or irrational. A constant is any number with a fixed value.
Simplify the inequality: (Enter your answer with the x variable on the left) 3(x−2)<4(x+2)
x > -14 is Correct × Correct. Begin by distributing on both sides of the inequality. 3x−6<4x+8 Now move the constants to one side and the x terms to the other side. 3x−3x−6<4x−3x+8 −6<x+8 −6−8<x −14<x Reverse the inequality to put the variable first. x>−14
Write the following equation in slope-intercept form: y=2(x+4)
y=2x + 8
Write the following equation in slope-intercept form: −9x+y−1=0
y=9x+1
In the inequality y≤−1.5x+4 , when x=−3 , y can equal ANY of which set of numbers?
−9 , −1 , 0 , 7 , 8.5 Correct. When x=−3, the inequality simplifies to y≤−1.5(−3)+4, so y≤8.5. All those numbers are less than or equal to 8.5.
Examples of Linear Equations:
✔ y=8 ✔ y=4x+10 ✔ −6+y=−2x ✔ y3=5 ✔ x+9=2y
A line has the coordinates of (2,5) and (3,−5) . Will the slope go uphill or downhill
. Downhill. Using the formula rise over run, (−5−5)(3−2)=−10. Therefore, the slope is negative, indicating that the line will go downhill.
Using the formula rise over run, what is the rise of a line that has the following two coordinates: (0,0) and (3,6) ?
. 6 is Correct × Correct. 6. The rise is y2−y1. If y2=6 and y1=0, then the rise is 6−0=6.
The following inequality would have solutions in which quadrants? y≥2x+7
. I, II, and III only Correct. The y-intercept is 7 and the slope is positive, and the shaded side of the line will pass through quadrants I, II, and III, but not IV.
In the inequality y>2x+5 , when x=−3 , y can equal which of the following values? (There is more than one correct answer).
1 Correct. y>2x+5 means that y>2(−3)+5=−1, and 1>−1.
(7−14s−21t)/7
1 - 2s - 3t is Correct × The answer is 1−2s−3t. Alter the fraction to reflect a multiplication operation, and with the new multiplication operation, we can use distributive property and simplify the expression. Therefore,
What is the y -intercept of the following linear equation: y=2(2x+5)
10 resents slope. 10 is Correct × Correct. 10. If we manipulate the equation so that it is written in slope-intercept form, we find that the equation is y=4x+10. In slope-intercept form, (y=mx+b), b represents slope.
What is the slope of the following linear equation: −4x+2y−10=0
2
What is the slope of a line that has the following two coordinates: (0,0) and (3,6) ?
2 is Correct × Correct. 2. The slope is rise (y2−y1) over run (x2−x1). Therefore, (6−0)(3−0) simplifies to 6/3=2.
Which of the following below is NOT an equation?
45y4−13y3+10y+4 Correct. The answer is c. Because there is no equals sign, this is a mathematical expression, rather than a mathematical equation.
Eliminating the Negative Sign
4=35−d Solve for d . −d can also be written as −1⋅d. So, to undo this multiplication (−1⋅), we divide −d by −1. By the division principle of equality, we must perform this division on both sides of the equation:
What is the y-intercept of the following linear equation: y=26x+6
6
3a−8a+2b(4a+1)
8ab - 5a + 2b is Correct × The answer is 8ab−5a+2b. First, combine your Like Terms: 3a and −8a. Then, distribute the 2b to the 4a and to the 1 inside the parenthesis. Our final answer is 8ab−5a+2b.
What is the y-intercept of the following linear equation: y=−39x+96
96
Graphing on the Coordinate Plane
A graph, or coordinate plane*, consists of an x-axis* and a y-axis*; the x -axis is the horizontal line* that passes through the point where y=0 , while the y -axis is the vertical line* that passes through the point where x=0 . We understand the points taken together to make up a graph. The graph can be the visual display of an algebraic equation.
A linear equation*
A linear equation* is an equation that has a degree* of 1 . When drawn on a graph, a linear equation creates a straight line (hence the name linear). Essentially, any equation that does not contain square roots* or exponents* greater than 1 is a linear equation. Below are a few different examples of graphed linear equations.
In Writing
A parenthesis, ( , denotes that the value alongside it is not to be included (as in < and > ). A bracket, ] , means that the value alongside it is included (as in ≤ and ≥ ). Use a comma between the two values to represent the interval.
Terms
A term* can be many things: a single constant, such as 5 , a single variable, such as x , or a term can also be any number of constants and variables multiplied together, such as 7ab . So, terms can include: Multiplication Constants, which can be coefficients Exponents Variables
Which of the following is considered an arithmetic expression: a. −4+−4 b. 2−1×5 c. 5×(7/3)−9 d. −2×(−3+9−2)+5 e. All of the above
All of the above are arithmetic expressions. They are each a string of numbers connected by elementary operations.
On a Number Line
An open circle denotes that the value is not to be included (as in < and > ). A closed circle means that the value is included (as in ≤ and ≥ ). Because an inequality is being represented, the graph drawn on the number line will represent a section of the number line, rather than a point.
Calculate the slope of a line that contains the following two coordinates: (−ab,c) , (ab,3c) .
Correct. The answer is b. Using the rise over run formula, we plug in the values of these coordinates to read as follows: 3c−cab−(−ab)=2c/2ab , which simplifies to c/ab .
If the rise =18 and the run =−3 , what is the slope of the line?
Correct. −6. If the rise =18 and the run =−3, then 18/−3=−6. The slope is -6
For y=5x−6 , the y -intercept is equal to −6 , which is plotted with a green point.
For y=5x−6 , the slope is equal to 5 , or 51 . From the −6 -intercept, travel up 5 and over 1 . The first point is at (1,−1) ; the next at (2,4) and so on.
Slope-Intercept Form
Knowledge of the coordinate plane allows us to graph algebraic equations. The most basic to graph are those that generate a straight line — also known as linear equations.`
x-Intercepts
Like the y -intercept, the x -intercept is the point at which the line crosses the x -axis. The y -intercept is given to us in the standard formula y=m+b . If you substitute 0 in for x , then y=b . This is the point at which x=0 (the y -axis) and the point along the y -axis. To find the x -intercept, substitute 0 in for y . For example, in the equation y=x−3 , when y=0 , we have 0=x−3 . To solve for x , add 3 to both sides. x=3 . The line y=x−3 crosses the x axis at the point (3,0) .
Direction of the Inequality
Much of the time, we manipulate inequalities the same way we would manipulate equations. We can simplify any term or expression on either side of the inequality. We can add or subtract values from both sides. We can multiply or divide both sides by a positive number. There are actions, however, that affect inequalities in unique ways. Multiplying or dividing both sides by a negative number will cause the direction of an inequality to reverse. Examine the table below to determine the reverse of each inequality sign: Reverse Inequality Signs > becomes < if we multiply or divide by a negative number ≥ becomes ≤ if we multiply or divide by a negative number < becomes > if we multiply or divide by a negative number ≤ becomes ≥ if we multiply or divide by a negative number To understand why this happens, see the following example: −6<−4 . We can see this represented on the number line below:
Strict Inequality
Plugging in solutions is actually a good way to verify that you have shaded the correct portion of the graph. If a point satisfies the inequality (that is, leads to a true statement, such as 5<19 ), then it needs to be in the shaded region. All sets of coordinate points that satisfy the inequality will be in the shaded region; all that don't will be in the unshaded region. When an inequality is strict, that is, either > or < , then the line is dotted. For example, consider the function y<12x+2 . The graph of this inequality is shown below.
We solve double inequalities just as we do simple inequalities, manipulating all sides of the inequalities, and remembering to switch the directions of both inequalities if multiplying or dividing by a negative number.
Solve the following inequality: −3≤2x+5<11 Subtract 5 on every side of the inequality: −3−5≤2x+5−5<11−5 −8≤2x<6 Now divide every side of the inequality by 2 . Since 2 is positive, we do not change the direction of the inequalities: −8/2≤2x/2<6/2 which becomes −4≤x<3 On a number line we represent this inequality as:
Solve the following inequality: −5y≥−30x+30
Solve the inequality above by isolating y on one side of the inequality symbol. Remember to perform the same operations to each side of the inequality symbol −5y≥−30x+30 y≤(−30x+30)÷−5 Notice that in the step above, the direction of the inequality reversed from ≥ to ≤ . This is due to the fact that we divided both sides by a negative number. Whenever an inequality is divided by a negative number, the direction of the sign is reversed. y≤6x−6 Above, we isolated y by dividing the left side by −5 and the entire right side by −5 . Since the solution above is in slope-intercept form, we can now graph the inequality with a line that has a slope of 6 and a y -intercept of −6 . Because the inequality is less than or equal to, the line drawn will be solid to indicate that the line itself is included in the solution. It is included in the solution because of the word "equal" in less than or equal to. Because y is less than or equal to, all of the area below the line y=6x−6 will be shaded in.
4x−4<2(6x+14)
Solve the inequality above by simplifying through distribution, manipulating both sides with addition and subtraction, and finally using division to isolate x . 4x−4<2(6x+14) Distribute the 2 to both terms. 4x−4<12x+28 Now subtract 4x from both sides. −4<8x+28 Subtract 28 from both sides. −32<8x Divide both sides by 8 . −4<x When we solved this problem, we were careful to avoid dividing by a negative number. If your instinct was to solve it a different way then you will likely need to change the direction of the inequality! The inequality −4<x can be rewritten as x>−4 , so our variable is on the left (which is often preferable).
Inequalities with One Variable
Solving an inequality with just one variable is like solving any other algebraic equation— with one important exception. Manipulate both sides of the inequality to isolate the variable and find the solution. Whenever you multiply or divide both sides of an inequality by a negative number you must switch the direction of the inequality.
Graphing Linear Equations
The first step in drawing the linear equation is plotting the y -intercept. Y -intercepts can be negative or positive, but they will always dictate where the line for the linear equation crosses the y -axis. The y -intercept for the following equation is 2 , which is plotted with a green point on the y -axis.
Principles of Equality
The principle of equality exists for each of the elementary operations* (addition, subtraction, multiplication, and division) and they are used to solve equations. We will study this in much more depth on subsequent pages, but to get a basic idea, let's start with some linear equations: x−3=4 If you subtract 3 from a number, you get 4 . You can probably do this problem in your head, but let's see how the principle of equality helps us solve it. If we add the same number to both sides, that gives us an equivalent equation: x−3=4 +3 +3 x−3+3=4+3 Notice that −3+3=0 x−3+3=4+3 x=7
To graph y=−0.5x−5 , first graph the y -intercept, −5 .
The slope for y=−0.5x−5 is equal to −0.5 . From the −5 y -intercept, travel down 1 and over 2 . The first point is at (2,−6) ; the next at (4,−7) and so on.
Inequalities with Two Variables
These types of inequalities, however, are solved graphically rather than just numerically. For example, an equation reads y≥2x+5 . Because the inequality symbol is greater than or equal to, we can interpret the inequality in two ways: y=2x+5 or y>2x+5 . Input values for x to find what values y is either greater than or equal to. For example, if x=−2 , then it follows that y≥2(−2)+5 = 1 , so y≥1 . Graphing inequalities is a large part of understanding their characteristics, and because they do not have defined values, rather than just graphing a line like in a linear function, graphed linear equalities incorporate both plotted points and shaded regions.
Solve the following inequality: −2≥−2x/3≥−4
To eliminate the denominator, we start by multiplying each side of the inequalities by 3 : −6≥−2x≥−12 Now divide each side by −2 . Since this is a negative number, we need to reverse every inequality: 3≤x≤6 On a number line we represent this inequality as:
Checking Solutions
We can also work with x and y coordinates to determine elements of two-variable inequalities. It is important to note that checking must be done with the initial inequality! So instead of plugging points into the simplified inequality, use the original one. For example, a plotted point on the coordinate plane sits at (2,7) ; as in x=2 and y=7 . Determine whether that ordered pair is a solution for the inequality −5y≥−30x+30 . Checking Solutions Example x=2, y=7 −5y≥−30x+30 −5(7)≥?−30(2)+30 −35≥−30 This is a not solution, as the above inequality is FALSE. −35 is not greater than or equal to −30 . After plugging in the value for x and y , we were able to find that the ordered pair (2,7) is NOT a solution for the inequality. Try (2,−4) with the same inequality: x=2, y=−4 −5y≥−30x+30 −5(−4)≥?−30(2)+30 20≥−30 This IS a solution, as the above inequality is true.
Inequality Notation
When working with inequalities, there's a number of different ways we can write and interpret their values. Take a look at the chart below. The first row displays the ways less than and greater than inequalities appear both in writing and when drawn on a number line. The second row shows how less than or equal to and greater than or equal to intervals appear. For example, what other ways can we write and display x≤2 ?
Non-Linear Equations
b2=2 c=d−−√ 7=D22 Notice that a linear equation can have terms in which a variable is multiplied, or divided, by a constant. These terms can then be added or subtracted from one another. However, a linear equation cannot have a term in which variables have exponents larger than 1 . Notice, too, that an algebraic equation in which a root of a variable is taken is also considered non-linear.
Coefficients and Exponents
coefficient* is a constant* written in front of a variable. A coefficient is written next to, specifically in front of, a variable because it is being multiplied by that variable. In a term, an exponent* is written as a superscript, above and to the right of a variable. For example, 3y is equivalent to y+y+y .
Which of the graphs below shows y<−3x+2 (Enter the letter that corresponds with your answer)
d is Correct × −3x+2 has a y-intercept of positive 2, which means it goes through the point (0,2). It has a slope of −3, meaning for every one unit it goes to the right the line goes down 3 units. Finally the less than symbol means the line should be dotted, not solid.
the distributive property.
distributive property* is a mathematical principle that is used to multiply a single term by multiple terms:
What is the slope of a line that has the following two coordinates: (−7,2) and (−5,3) ?
ect. .5. The rise is 3−2=1. The run is −5−(−7)=2. Therefore the equation becomes 1/2=1/2 or .5.
Linear Inequalities
large part of math and of algebra involves using the basic tools and concepts together to solve even more complex problems. Having learned inequalities and linear functions, we can now combine the two to understand and solve linear inequalities. A linear inequality works similarly to basic linear equations, but incorporates inequality signs, like < , > , ≤ , and ≥ , rather than equality signs ( = ).
variables*
of mathematics. In elementary algebra, a variable is a symbol that represents or holds the place of a numerical value*. Often, a variable will be a letter from the Western alphabet ( a , b , c , ...) or Greek alphabet (α, β, γ, ...). The actual letter or symbol being used as a variable is not important. The numerical value represented by the symbol is what gives the variable its importance.
Which of the following inequalities matches the graph below? (Enter the letter that corresponds with your answer.)
rrect. The slope is −2/5. You can determine this by noticing that the line goes down 2 for every 5 units it goes to the right. Or you can use the points (0,1) and (5,−1) to determine the slope: −1−15−0. The y-intercept is 1. The symbol is ">" because the line is dotted and the shading is above the line.
Slope
slope of a line is its pitch. A large coefficient in front of the x creates a steep line while a small coefficient creates a gentle slope. Positive slopes create "uphill" slopes: imagine walking from left to right and you will ascend. Compare the two positive slopes below:
the Butterfly Method
the Butterfly Method* is a way to cross-multiply two fractions to determine whether they are equal. We can also use the Butterfly Method to cross-multiply and solve algebraic equations of the form: (a/b)x=c/d , where a , b , c , and d are constants and x is a variable.
combining Like Terms
we can only combine like terms* using addition and subtraction. While it makes sense to add one jar to two jars, leaving you with three jars, this concept is a bit more intricate with multiplication and division. For now, remember that like terms* can only be combined using addition and subtraction.
Double Inequalities
we have inequalities that say that the variable is between two values. These inequalities we call "double inequalities". For example, say that x is greater than −2 , but less than 4 . In separate inequalities, this would become: x>−2 and x<4 which we can write as a double inequality: −2<x<4 On a number line we represent this as :
Simplify the inequality: (Enter your answer with the x variable on the left) fraction begin. numerator: x + 5 denominator: 3 fraction end. < fraction begin. numerator: x - 7 denominator: -4 fraction end.
x < 1/7 is Correct × Correct. Since x+5 is divided by 3 multiply both sides by 3: 3⋅x+5/3<3⋅x−7−4 x+5<3x−21−4 Multiply both sides by −4 and reverse the sign −4(x+5)>3x−21 −4x−20>3x−21 −7x>−1 −7x−7<−1−7 x<17
A line has a slope of −3 and passes through the point (5,2) . What is the equation of the line in slope-intercept form?
y=−3x+17 Correct. The answer is c. We can find the slope-intercept form of this line by using the equation y=mx+b. We know this line has a slope of −3 and passes through the point (5,2), so we can plug those values into the equation and solve for b:
17w−9=4w+19 17w−4=4w+24 Compare the second statement to the first. Does the second show the principle of equality? Write "yes" if it does; "no" if it does not.
yes is Correct × The answer is yes. You can see that 5 has been added to the right side of the equation because −9 became −4. 19 became 24 on the right side of the equation, so 5 was added. Since 5 was added to both sides of the equation, the principle of equality was upheld.
What is the slope of the following linear equation: y=−98x+16
-98 −98. This equation is in slope-intercept form, which is y=mx+b, where m represents slope
9. What is the y -intercept of the following linear equation: y−10x=0
0
4x2+5x(2x+8y−3y)=
14x^2 + 25xy is Correct
Division Principle of Equality
2x=10 Solve for x . We want to solve for x . On the left side of the equation is a multiplication operation: 2⋅x . Therefore, we need to undo the multiplication operation ⋅x . Multiplication and division are inverse operations*, so we can use division to undo the multiplication here and isolate x . By the division principle of equality, we divide both sides of the equation by 2 : 2x=10÷2 2x÷2=10÷2 x=5
(9w−3)/(4.5w+12−4.5w−9)
3w - 1 is Correct × The answer is 3w−1. First, combine like terms in the denominator. The 4.5w and the −4.5w will cancel out, and combining the constant terms gives 12−9=3. Then, alter the fraction to reflect a multiplication and distribute. Therefore,
Addition is the inverse of subtraction
5 − 2 = 3 3 + 2 = 5 his process only works for nonzero real numbers. This process fails for multiplication and division if the number zero is introduced.
Combining Like Terms with Subtraction
Always use the coefficients of the term to combine like terms. Regardless of whether addition or subtraction is being used, like terms are combined by performing the operation on the coefficients of the terms. For example, examine the expression above, 9y − 3y . These like terms can be combined by operating upon the coefficients:
Linear Equations
An algebraic equation* is any equation that contains variables*, constants*, or mathematical operations*. A linear equation* is an algebraic equation with degree* of 1 . This means that none of the terms* in the equation have an exponent larger than 1 . For example, examine the algebraic equations below. They are classified as either linear equations or non-linear equations below: Linear Equations b=2+x b=d/2 7=8D
Algebraic Expressions
An algebraic expression* is a string of terms connected by division, addition, and subtraction. Consider the expression below. We recognize the individual terms that are separated from each other by division, addition and subtraction. These terms make up an algebraic expression.
Arithmetic Expressions
An arithmetic expression* is a string of numbers connected by elementary operations. Arithmetic expressions are the addition, subtraction, multiplication, and division problems you are probably familiar with. These expressions can be of any length. They can be short and simple:
Equations
An equation*, denoted by the equals sign, =, represents a mathematical relationship. Following from it's name, equations denote that entities are equal* to one another. More specifically, equations represent that two expressions*, lying on opposite sides of an equals sign, hold the same value as each other. Examine the following equation: a=3
You need to write an expression that adds together the terms 8X2 , 2Y4 , 3XY3 , and 27 . What is the correct order for the terms to show in the final expression?
he answer is d. 2Y4+3XY3+8X2+27 . Always write expressions in descending order of exponent value with constants at the end of the expression. In the given terms, the term 2Y4 has the largest exponent, so it is written first, with lower exponents following in order. The constant 27 comes last. Therefore, the answer is 2Y4+3XY3+8X2+27 .
3y=13z 3y−12=13z−12 Compare the second statement to the first. Does the second show the principle of equality? Write "yes" if it does; "no" if it does not.
is Correct × The answer is yes. 12 has been subracted from both sides of the equation 3y=13z, so the principle of equality has been upheld.
An equation*,
An equation*, fundamentally, is simply two terms or expressions with an equals sign in between them. In algebra, we are often manipulating these terms or expressions by adding, multiplying, or performing other operations. For example, y=−8 ; this is considered the simplest form of an equation. This type of equation, in which the value of variable is found, is also your goal in solving algebraic equations; once you've found the value of variable, you've essentially found the solution. It is important to remember that both sides of any equation are equal to one another.
Exponents
An exponent*, sometimes called a power, is a quantity that represents repeated multiplication. An exponent is written above and to the right of a number, known as a base. The exponent's value is equal to the number of times the base is multiplied by itself. To the right, the base is 6 and the exponent 3 . 63 is equal to 6×6×6
Operations
An operation* is a mathematical procedure which can generate a new value. Elementary operations* are the simplest and most common operations: addition, subtraction, multiplication, and division.
Inverse Operations
An operation* is a procedure which generates an output from one or more inputs. The four most common, basic elementary operations* are addition, subtraction, multiplication, and division. Two operations are considered inverse operations* if they undo one another. For example, addition is the inverse of subtraction. Examine the image below:
In the expression 32×5÷1−2 , which value is a base? 3 2 5 1
Correct. The answer is a. The 3 is the base in the exponent 32.
What elementary operation will you use to generate the value 7 from the constants 3 and 4 ? Addition Subtraction Multiplication Division
Correct. The answer is a. The constants 3 and 4 can generate the new value 7 when they are added. Therefore the answer is addition.
Which of the below are constants? Select the most correct answer. 3.2 3/7 −0.0009 All of the above
Correct. The answer is d. A constant is a number with a fixed value. All real numbers are constants, regardless of if they are prime or composite, odd or even, rational or irrational. Therefore, since all the above numbers are fixed, they are all constants.
Which of the following operations are elementary operations? Only addition and subtraction Only multiplication and division Only addition and multiplication Addition, subtraction, multiplication, and division
Correct. The answer is d. Elementary operations are addition, subtraction, multiplication, and division.
Exponents
Exponents How about operations like a · a that involve two of the same variables being multiplied? You may recognize this operation as repeated multiplication. This type of operation is best represented using exponents. Exponents* are used for repeated multiplication of a variable, the same as with repeated multiplication of a constant. a⋅a = a2
Expression Classification
Expressions are often classified by their degree. The following list shows the names of expressions based on their degree: Names of Expressions Based on Their Degree An expression of degree 0 is known as a constant* An expression of degree 1 is known as linear* An expression of degree 2 is known as quadratic* An expression of degree 3 is known as cubic
Getting the Equation into Slope-Intercept Form
First, make sure the y term is on the left side of the equation. Put all x 's and constants on the right side of the equation. Multiply or divide to make the coefficient of y be 1 .
Distribution of Negative Numbers
For example, the expression 3x(5y−2) can be simplified using distributive property. However, with the inclusion of a subtraction operation, properly translating the expression can become tricky. When in doubt, change all subtraction operations to the addition of negative numbers.
Identifying Like Terms
For terms to be like terms*, they need to have the same variable(s) with the same exponent(s). When identifying whether two terms are like terms, we only need to examine the variable(s) and their exponent(s); we can ignore the coefficient in the term.
Calculating Slope
How is slope actually calculated? The slope is a proportion. It is a consistent ratio of distance along the y -axis to distance along the x -axis. Consider the equation y=5x . The coefficient 5 in front of the x (the " m " in y=mx+b ), tells us that for every additional 1 of x -value, the y -value will increase by 5 . When x=1 , y=5 . When x=2 , y=10 . The difference between the initial y -value of 5 and the second y -value of 10 is 5 . As x increases by 1 , y increases by 5 . We can see this graphically as well:
The multiplication principle of equality*
If both sides of an equation are multiplied by the same non-zero number, the result is an equivalent equation.
positive and negative quadrants
Quadrant I= positive x -coordinate, positive y -coordinate Quadrant II= negative x -coordinate, positive y -coordinate Quadrant III= negative x -coordinate, negative y -coordinate Quadrant IV= positive x -coordinate, negative y -coordinate
slope* and the y-intercept*.
Slope-intercept form is made up of two unique parts: the slope* and the y-intercept*. The slope, or m , represents the steepness of the line drawn on a graph. The y -intercept, or b , determines where on the y -axis the line crosses.
Solve the following algebraic equation: 23/21j=25/13 .
Step 1 Write the algebraic equation in the form ax/b=c/d: Remember that a/b x=ax/b since x can be converted into x1 Now draw "butterfly wings" around the opposite terms. In this example, group 23 and 13 , and 21 and 25 , like this:
3x+3/4y=−4x−7 into slop intercept
This equation is not in slope-intercept form. We have to manipulate it. Get the x 's all on the right side. 3x−3x+3/4y=−4x−3x−734y=−7x−7 Multiply both sides by the reciprocal of 34 : 43(34y)=43(−7x−7)y=4⋅−73x−4⋅73y=−283x−283y=−9-1/3x−913 Now, we have an equation we can easily graph; our slope is −9-1/3 and our y -intercept −9-1/3 .
net margin.
To calculate net margin, all expenses must be subtracted from revenue, or total money taken in, which will equal net income. Then, net income is divided by revenue to accurately find the remaining value. Net income=Revenue−(Cost of goods sold+Operating expenses+fixed costs) Net margin=Net Income÷Revenue
Unlike Terms
Unlike Terms: a and b Here, the terms have two different variables: a and b . They are NOT like terms. Unlike Terms: 7x and 7y Don't be confused by the matching coefficients! Here, the terms have two different variables once coefficients are removed: x and y . The variables do not match, so these are NOT like terms. Unlike Terms: 3m and 7m2 Be careful! Here, the terms have the same variable, m, but with different exponents. Once coefficients are removed, we are left with m and m2 . While these may appear similar, the fact that the variables have different exponents ensures that they are NOT like terms. Remember: like terms always have identical variables with identical exponents. Terms that have similar, but slightly different variables are not like terms and cannot be combined. Similarly, terms that have the same variables, but slightly different exponents on those variable are not like terms and cannot be combined.
Find the slope of the line that contains the points (1,5) and (0,−3)
Using the two points we are given, we substitute them into the formula −3−5/0−1 The slope equals −8/−1 or 8
Combining Like Terms with Addition
We recognize that these are like terms by ignoring their coefficients. If we imaginatively remove their coefficients, these terms have the same variable: h . How do we combine these like terms? That's when the coefficients come back into play. Like terms can be combined by adding the coefficients of the terms. Take the expression 4h+7h . The coefficients of the like terms are 4 and 7 . We recognize that these are like terms by ignoring their coefficients. If we imaginatively remove their coefficients, these terms have the same variable: h .
principle of equality
If x+4=10 , then what does x equal? To solve for x , we must manipulate the equation so that x stands alone on one side of the equals sign. To do this, we need to undo the x on the left side of the equation. We know, by inverse operations, that subtraction undoes addition. So, we can simply subtract 4 from the left side of the equation to isolate x . But, we have to perform the same operation to both sides of the equation. So, using the subtraction principle of equality*, we subtract 4 from both sides of the equation! x+4=10 −4 −4x=10−4x=6
The principle of equality*
If you perform equivalent operations to both sides of an equation, the result will always be an equivalent equation. 4=4
Multi-Step Expressions
In Multi-Step Expressions We must first evaluate the expressions containing known values to find the value of y. (7⋅2)−6+y=18(14)−6+y=188+y=18 Now that we've evaluated the expression on the left side of the equation, we can find y 's value.
Why do we use the distributive property?
It is evident that the method on the left (using distribution) has one more step than the process on the right. There is an extra step on the left because we are not combining like terms ( 4 + 3 ) before performing any other operations, like happens when using the method on the right (order of operations). Often, the terms inside of the parentheses are not like terms and therefore cannot be combined. Then, the distributive property will be the only way to simplify the expression. When there are not like terms inside of the parentheses, or once you have finished combining all possible like terms, distribution is the method you'll need to employ to simplify the expression.
Like Terms
Like Terms a and 2a −4 and 7 3D2 ; and 6D2 The terms a and 2a are like terms because they share the same variable: a . After ignoring any coefficients in the terms a and 2a , we are left with the same variable: a . There is no exponent written next to either variable, so the exponent is the same for both terms. The terms −4 and 7 are like terms because they are both constants*, and therefore they both lack a variable. This lack of a variable constitutes an identical variable. The terms D2 and 6D2 are like terms because they share the same variable ( D ) raised to the same power ( 2 ). For both terms, the "variable and its exponent" is D2 .
dosage
Medication dosages may be figured using a proportion ratio formula such as: Dosage on hand/Amount on hand=Dosage desired/x(Amount desired) Or they may use: Desired amoun/tAmount on hand⋅(Volume)=x(Amount to give)
Why Multiplication is Unique
Multiplication is the only elementary operation* that can be denoted by variables and constants simply being written next to one another: 3⋅x can be written as 3x . In other words 3⋅x and 3x mean the same thing. Similarly, −6⋅a can be written as −6a , and 14⋅d can be written as 14d .
Distributive Property with Exponents
One final consideration when using the distributive property is the presence of exponents*. Often there will also be instances in which you're asked to multiply the same variables by one another. For example, an expression that reads 3x(x) , would be simplified by multiplying 3x by x . The simplified expression would be written as 3x2 . Because we're multiplying a variable by itself, the product results in an exponent. As we've seen elsewhere, the term 3x2 can be broken up as 3⋅x⋅x . This concept is seen regularly when simplifying algebraic expressions. Example: We must use distributive property to simplify the expression below: 2x(8y+4x) .
Creating Terms: Multiplication
Operations* work similarly with variables and constants. Multiplication generates a product: a completely different operation than addition or subtraction. Take 2⋅6 for example. If we add 2 and 6 , the sum is 8 ; but when multiplying 2 and 6 , the product is 12 . We're connecting these constants using multiplication in a different way than addition. As with numbers, multiplication of variables is a wholly unique operation. With multiplication, these variables form a product. Our product is a⋅x .
points plotted are called
Points plotted on a graph are located via coordinates. The location on the x -axis is written first, and the location on the y -axis second — the x -coordinate followed by the y -coordinate. These two coordinates — also referred to as ordered pairs* — tell us the locations of each point and in which quadrant* they fall. All four quadrants meet at the point (0,0) . This point is known as the origin*. The origin is also the point at which the x -axis and y -axis intersect.
Steps for Solving an Equation With Complex Expressions
Substitute any variable's known value for the variable itself Simplify expressions on either side of the equation following order of operations: Distribute Combine Like terms Add and subtract constants Complete any other process that serves to simplify the expression Move terms across the equation, using the Addition and Subtraction Principles of Equality: Move all constants to one side of the equation Get all terms with the variable to be solved on the opposite side of the equation Simplify the expressions on either side of the equation: Combine like terms on one side, if necessary Add and subtract constants on the other side, if necessary Isolate the lone variable on one side of the equation, using the Multiplication and Division Principles of Equality: The variable will be across from its value Check your answer: Plug in your solution to the original equation. Perform the arithmetic on both sides of the equation. If the two sides of the equation are equal, you have successfully solved the equation!
Each of the elementary operations have an inverse operation:
Subtraction is the inverse of addition. Addition is the inverse of subtraction. Division is the inverse of multiplication. Multiplication is the inverse of division.
Manipulating Equations with Addition and Subtraction
The act of manipulating equations is often for the purpose of solving for a variable*. We must first understand and learn the steps needed to find that final product in order to move forward. Because addition and subtraction are inverse operations*, they undo one another. We can use this fact when solving for a variable in an algebraic expression. The key is to manipulate the equation so that the variable is isolated on one side of the equals sign—that's how we can accurately solve for the variable's value. Equations are like mathematical scales—in order for them to be balanced, we must perform the same operations on each side of the equation.
degree*
The degree* of an expression refers to the largest exponent in an expression. 3x2 Degree =2 For this expression, the degree is 2 because the exponent is 2 . Example #2 7a Degree =1 For this expression, the degree is 1 because the implied exponent is 1 : 7a=7a1 Example #3 9m4−2z2 Degree =4 In this expression, m has an exponent of 4 and z has an exponent of 2 . The degree of an expression is equal to the largest exponent, so the degree here is 4 . 12 Degree =0 For this expression, the degree is 0 because the constant can be imagined as having an implied variable with an exponent of 0 : 12=12x0
How the Distributive Property Works
The distributive property uses a process known as distribution. We can essentially distribute the term outside the parentheses to each of the terms inside the parentheses. To see how this process works, let's first use distribution with constants. 2(4+3) there are two different ways to evaluate this expression: the distributive property or the regular order of operations* that we know. In the diagram below, the left column uses the distributive property to evaluate the expression. The right column evaluates the expression without using the distributive property.
−6+y=−2x put in slop intercept form
The example above is not written in slope-intercept form, however we can manipulate the equation algebraically. −6+y=−2x +6 +6 ✔ y=−2x+6 Now, we have an equation we can easily graph; our slope is −2 and our y -intercept 6 .
Manipulating Equations With Multiplication and Division
The rule of performing the same operations to each side of the equation also applies to equations involving multiplication and division. Like the addition and subtraction principles of equality, there are also multiplication and division principles of equality:
ow many terms are in the following expression: 3b−4c
The terms in the expression are 3b and 4c . A term can also be any number of constants and variables multiplied together. Here, subtraction separates the two terms from one another.
Constants
The word constant may seem intimidating at first, but you work with constants all of the time. A constant* is a number with a fixed value. All real numbers are constants, including 0 , 1.5 , −10 , and π . Any of the numbers we used in "Number Systems" are constants, regardless of whether they are prime or composite, odd or even, rational or irrational!
x+8=2y put into slop intercept form
This equation is not in slope-intercept form. We have to manipulate it. First reverse the equation to get the y on the left side. 2y=x+8 Divide both sides by 2 2y/2=x+8/2 y=x/2+8/2 Simplify the slope and intercepts as much as possible. y=1/2x+4 Now, we have an equation we can easily graph; our slope is 1/2 and our y -intercept 4 .
Substitution
When we know the value that a variable represents, we can simply replace the variable with its value. Then, we can evaluate the expression. For example, if we know that variable a represents a value of 1 , we can say that a equals 1 . Because we know that the value of a is 1 , we can replace the variable a with the number 1 in an expression. This is outlined in the examples below: Variable Value a 1 Using the table above, we have a variable, a , which represents the numerical value of 1 . Therefore, the statement (a+5) is equivalent to (1+5) . Because the variable a represents the numerical value of 1 , we can simply replace the variable a with the number 1 .
Slope-Intercept Form working
When working to plot linear equations, it is crucial to be familiar with slope-intercept form*. Slope-intercept is a common format that a linear equation can take that is helpful for graphing purposes. It is of the following form: y=mx+b , where m is the slope of the line and b is the y -intercept.
Writing Algebraic Expressions
When writing an algebraic expression, you want to consider the order in which you write the terms that make up the expression. Write the constants at the end of your expression. 6x+11 Notice that the expression above has two terms: 6x and 11 . One of these terms ( 11 ) is a constant. Because this term is a constant, we place it at the end of the expression. Here are some more examples of expressions with constants placed at the end of the expression: y3−51 abc+5 x+8−−√ Note: We have seen that within a term, constants are written in front of variables. These constants are known as coefficients. So, while constants are written at the end of expressions, they are written at the beginning of terms. This can be a point of confusion, and practice makes perfect with writing these terms and expressions! Write the terms with the largest exponents first in your expression. x3−8b2 This expression has two terms: x3 and 8b2 . Since x3 has an exponent of 3 and 8b2 has an exponent of 2 , we write x3 first in the expression because it has a larger exponent. Always write expressions in descending order of exponent value with constants at the end of the expression. Here are some examples:
There is a formula that captures this ratio which you should use to calculate the slope when given two points
When you have two points (x1, y1) and (x2, y2) : Slope = Rise: (y2−y1)/Run: (x2−x1)
Evaluate the following expression using substitution, given that g is −4 . (g+1)(g+2)(g+3)
a. −6 Correct. The answer is a. Replace g with −4 and then perform the operations. Remember the order of operations. Therefore, (g+1)(g+2)(g+3)= ((−4)+1)((−4)+2)((−4)+3)= (−3)(−2)(−1)=−6.
Multiplication Principle of Equality
a÷4=−3 ⋅4 ⋅4a=−3⋅4a=−12 a/4 = -3 4xa/4 = -3 x 4 a =-12
Coefficients
constants* often take the form of coefficients*. A coefficient is a number by which a variable is being multiplied. Coefficients are written in front of variables. So, in 16x , 16 is the coefficient and x is the variable. If a variable is without a number in front of it, the coefficient is 1 . Though it is not written, there is essentially an invisible 1 in front of any variable without a numerical coefficient.
*the substitution method*
n the expression above, " x+2 " does not have a known value*, because x does not have a specified, known value. If the value of x were specified, we could use the substitution method*: replacing x with its value. Substitution* is a method used in algebra when a variable is substituted by its known value. After performing substitution, we can evaluate the expression like any other. For example, say we know that x=3 . We can substitute the value 3 for the variable x . Therefore, the expression " x+2 " becomes " 3+2 ." It can easily be evaluated with a simple addition operation: 3+2=5 . Tip: when substituting a value for a variable, it is a good idea to enclose the value within parentheses. Using parentheses is a good visual way to keep track of values that have replaced variables. This process is outlined below:
