Algebra 1 Quarter 3 Project

8-1 Multiplying Monomials Constants

An expression involving the division of variables is not a monomial. Monomials that are real numbers are called constants The formula for the area of a square is s=4ab

8-4 Polynomials Binomial

Binomial is the sum of two monomials

8-6 multiplying a Polynomial by a Monomial Find -2x^2 (3x^2 - 7x + 10)

Find -2x^2 (3x^2 - 7x + 10) Use the vertical method 3x^2 - 7x + 10 -2x^2 Distribute = -6x^4 + 14x^3 - 20x^2

6-1 Solving Inequalities by Addition and Subtraction 1.If a>b then a+c>b+c 2. If a<b,then a+c<b+c

If any number is added to each side of a true inequality,the resulting in equity is also Example if 2<7 2+6<7+6 8<13 a more concise way to writing a solution set builder notation or {t|t_<_58}

6-5 solving open sentences involving absolute value |x|=n, |x|<n |x|>n

if |x|=n |x|=5 it means the distance between 0 and x is 5 units

5-3 Slope Intercept Form slope intercept form y=mx+b

is in slope intercept form EX. to write an equation of the line whos slope is 3 and whose y intercept is 5 y=3x+5 Towrite an equation when given 2 points you always write it as rise over run (x1,y1)=(0,3) and (x2,y2)=(2,-1)

8-8 special products Square of sum

square of sum (a+b)^2=a^2+2ab+b^2

7-2 substitution Substitution

the exact solution of a system of equations can be found by using algebraic methods. One such method is called substitution y=3x x+2y=21 to use substitution to solve the system of equations you need to know what the variables are equal to y=3x so you would substitute 3x in the second equation x+2y=21 x+2(3x)=21 then simplify x+6x=-21 and combine like terms 7x=-21 Divide each side by 7 _7x_=_-21_ 7 7 the you use y=3x to find the value of y y=3x first equation y=3(-3) x=-3 y=-9 The solution is (-3,-9)

5-7 Scatter Plots and Lines of Fit

the line that "explains" the trend of the data

7-4 elimination using multiplication Elimination using multiplication

Elimination using multiplication if none of the coefficients are 1 or -1 and neither of the variables can be eliminated by simply adding or subtracting the equations

8-7 Multiple polynomials (x+3)(x+2)

(x+3)(x+2) = X(x+2)+3(x+2) distributive prop =x(x)+x(2)+3(x)=3(2) Distributive Property =x^2+2x+3x+6 Multiply =x^2 +5x + 6 combine like terms

7-3 elimination using Addition and subtraction Write the equation in column form and add 3x-5y= -16 (+)2x+5y=31

1. Write the equation in column form and add 3x-5y= -16 (+)2x+5y=31 ---------------- 5x = 15 The variable 5x = 15 -- --- 5 5 x=3

7-4 elimination using multiplication 3x + 4y = 6 -10x-4y = 8

3x + 4y = 6 -10x-4y = 8 Add equations -7x=14 Then divide by -7 -7 -7 x= -2 simplify now you substitute -2 for x in either equation to find the value of y 3x+4y=6 First equation 3(-2)+4y=6 x=-2 -6+4y=6 Simplify -6+4y+6=6+6 Add 6 to each side 4y=12 Simplify _4_y=_12_ Divide each side by 4 4 4 y=3 The solution is (-2,3)

8-2 Dividing Monomials 4^5/4^3

4^5/4^3 subtract the exponents 5-3 =2 and the answer will be 4^2 or 4*4

8-3 scientific notation Example Problem

6.54 x 10^4 =6.59 x 10,000 = 65,900 the decimal point is moved 4 places to the right

8-2 Dividing Monomials 8^2/8^5

8^2/8^5 = 8^2-5 Quotient of powers =8^-3 Subtract for all integers m and any non zero number a, _a^m_=a^m-n a^n

8-4 Polynomials Polynomial

A polynomial is a monomial or a sum of monomial

7-1 Graphing systems of equations solution of a system of equations

A solution of a system of equations is an ordered pair of numbers that satisfies both equations The equation for solving system equations is y=x+n y=x-n for example y=-x+8 y=4x-7 in this case the graph intersects with (3,5) now replace the variables y=-x+8 y=4x-7 y=-3+8 5=^?4(3)-7 5=5 5=^?12-7 5=5 the solution is (3,5) and when graphed the lines a parallel

6-2 Solving inequalities by Multiplication and Division If a>b, then ac>bc and if a<b then ac<bc Division: If a>b then _a_<_b_ and if a<b then _a_>_b_ c c c c

If each side of an inequality is multiplied by a positive number, the inequality remains true. By multiplication 8>5 8(-2)_?_5(2) ,multiply each side by 2 16 > 10 Solving inequalities by division I14h>91 original inequality _14_ >_91_ 14 14 h>6.5

7-1 Graphing systems of equations Coincide

If the graphs are coincide, the system system of equations is said to be consistent. That is, it has at least one ordered pair that satisfies both equations. an example of a system of equations would be y=69-6.9x and y=5.7+6.3x they are an ordered pair of equations

8-1 Multiplying Monomials Monomial

Monomial Is a number, a variable, or a product of a number and one or more variables This is an example of a monomial _1_s^2 20 Monomials that are real numbers are called constants In x^n the base number is x and the exponent is n 2^5 is equal to 2x2x2x2x2 or 32

5-6 Parallel and Perpendicular Lines

Parallel Lines: in the same plane that never intersect are called parallel lines. If two nonvertical lines Parallel Lines have the same slope, then they are parallel. All vertical lines are parallel. Perpendicular Lines: Lines that intersect at right angles are called perpendicular lines. If the product of the slopes of Perpendicular two lines is - 1, then the lines are perpendicular. The slopes of two perpendicular lines are. Lines negative reciprocals of each other. In a plane, vertical lines and horizontal lines are perpendicular

8-8 special products product of a sum and a difference

Product of a sum and a difference (a-b)(a+b)=a^2-b^2

7-5 Graphing Systems of inequality Real Word Problems

Real World Problems In real-life problems involving systems of inequalities, sometimes only whole-number solutions make sense .

8-3 Scientific Notation Scientific Notation

Scientific Notation is when your dealing with very big numbers its an easy way to keep track of them

7-2 Substitution Use substitution to solve the system of equations. 2x - y = -3 x = 2y

Since x = 2y, substitute 2y for x in the first equation. 2x - y = -3 First equation 2(2y) - y = -3 x = 2y 4y - y = -3 Simplify. 3y = -3 Combine like terms then divide each side by three y=-1 then substitute -1 for Y in the equation x = 2y Second equation x = 2(-1) y = -1 x = -2 Simplify The solution is (-2, -1).

5-1 Slope rise change in quantity ---=----------------- run change in time

Slope is used to describe a rate of change, the rate of change tells average. or how much a quantity is changed over time. A number, variable, or a product of a number and one or more variables rate of change tells average how quantity is changing over time to determmine the slope of the line that passes through (0,-4) and (3,2) we are assigning (0,-4)=(x_1,y_1) and (3,2)=(x_2,y_2) slopeformula is m=_y2-y1_ x2 x1 m=_2-_(-4)_ 3 - 0 and simplify m=_6_=2 3

6-3 Solving Multi-step inequalities

Solving multi-step inequalities can be done by undoing the operations in the same way you would solve an equation with more than one operation For example -7b+19<-16 Original inequality -7b+19-19<-16-19 Subtract 19 from each side -7b<-35 Simplify _-7b_>_-35_ Divide each side by -7 and change <to> -7 -7 b>5 Simplify

5-5 Writing Equations in Point-Slope Form

Solving point slope form. whats the equation for the line that passes through (-1,5) with the slope -3. The equation will be the y-y1=m(x-x1) plug in the variables and its y-5=-3[x-(-1)] then simplify and you get y-5=-3(x+1)

7-3 Elimination using addition and subtraction Elimination

Sometimes adding two equations together will eliminate one variable. Using this step to solve a system of equations is called elimination the equation of elimination is n+d=v n-d=v then add vertically _n+d=v_ (+) n-d=v_ n =v

8-8 special products Square of difference

Square of difference (a-b)^2=a^2 - 2ab +b^2

8-5 Adding And subtracting Polynomials Example of subtracting polynomials

Subtract Polynomials (5a-6b) - (2a+5b) add the opposite (5a-6b) + (-2a-5b) =3a-11b

8-7 Multiplying Polynomials Foil Method

The foil method is a way of multiplying polynomials. First, outer, inner, last

6-6 Graphing Inequalities in Two Variables

The solution set of an inequality in two variables is the set of all ordered pairs that satisfy the inequality. Like a linear equation in two variable, the solution set is graphed on a coordinate plane. The solution set for an inequality is two variables contain many ordered pairs when the domain and range are the set of real numbers. The graphs of all of these ordered pairs fill a region on the coordinate plane called a *half-plane*. An equation defines the *boundary* or edge for each half-plane. 1.) Ordered Pairs that Satisfy an Inequality 2.) Graph an Inequality (Problem: Graph y-2x≤-4) (*Step One*: Solve for y in terms of x, original inequality, add 2x to each side, simplify. *Step 2*: Graph y=2x-4. Since y≤2x-4 means y<2x-4 or y=2x-4, the boundary is included in the solution set. The boundary should be drawn as a solid line. *Step 3*: Select a point in one of the half-planes and test it. Let's use (0,0). Since the statement is false, the half-plane containing the origin is not part of the solution. Shade the other half-plane.) (*CHECK*: Test a point in the other half-plane, for example, (3, -3). Since the statement is true, the half-plane containing (3,-3) should be shaded. The graph of the solution is correct.) 3.) Write and Solve an Inequality

6-5 Solving open sentences Involving Absolute value

There are three types of open sentences that can involve absolute value. |x|=n, |x|<n, |x|>n. Consider the case of |x|=n. |x|=5 means the distance between 0 and x is 5 units. If |x|=5, then x=-5 or x=5. The solution set is {-5,5}. When solving the equations that involve absolute value, there are two cases to consider. *Case 1*: The value inside the absolute value symbols is positive. *Case 2*: The value inside the absolute value symbols is negative. (Equations involving absolute value can be solved by graphing them on a number line or by writing them as a compound sentence and solving it). 1.)Solve an Absolute Value Equation 2.) Write an Absolute Value Equation 3.) Solve an Absolute Value Inequality (<) 4.) Solve an Absolute Value Inequality (>)

7-5 Graphing Systems of Inequalities Vocab

To solve system of inequalities you have to find the ordered pairs that satisfies all the inequalities y<-x+1 y_<_2x+3 the solution includes ordered pairs that satisfy both equations

8-4 Polynomials Trinomial

Trinomial is the sum of three monomials. Polynomials with more than three terms have no special names.

8-5 Adding And subtracting Polynomials Example of Vertical Method

Vertical Method Aline the like terms and add 3x^2 - 4x + 8 -7x^2 + 2x - 5 = -4x^2 - 2x + 3

5-4 Writing Equations Slope-intercept form

Write the equation of a line that passes through (1,5) with the slope 2 [y=mx+b] [5=2(1)+b] [5=2+b] then subtract 2 from both sides [5-2=2+b-2] [3=b] then plug in all the variables to the original equation which will be [y=2x+3]

6-4 Solving Compound Inequalities

a compound inequality is when two or more inequalities are taken A compound inequality containing *and* is true only if both inequalities are true. Thus, the graph of a compound inequality containing *and* is the *intersection* of the graphs of the two inequalities. In other words, the solution must be a solution of *both* inequalities. The intersection can be found by graphing each inequality and then determining where the graphs overlap. 1.) Graph an Intersection (Problem: Graph the solution set of x<3 and ≥-2) (Graph x<3, Graph x≥-2, find the intersection) (The solution set is {x|-2≤x<3}. Note that the graph of x≥-2 includes the point -2. The graph of x<3 does *not* include 3). 2. ) Solve and Graph an Intersection 3.) Write and Graph a Compound Inequality 4.) Graph and Solve a Union

5-2 Slope Direct Variation y=kx where k=0

direction variation is described by an equation of the form the Constant of the equation family of graphs:includes graphs and equations that have at least one characteristic in common the parent graph if the simplest graph To find the slope constant variation you must fin the slop of each graph. In this example the Constant for the first graph is (1,3) m=_3-_0_ 1-0 m=3 is the constant variation thenm we take our other graph (1,-2) and do the same thing m=_-2_0_ 1-0 m=-2 is the slope

8-6 multiplying a polynomial by a monomial y(y-12) +y(y+2)+25=2y(y+5)-15

y(y-12) +y(y+2)+25=2y(y+5)-15 y^2-12y + y^2+2y + 25 = 2y^2+10y-15 dist. prop 2y^2-10y+25 = 2y^2+10y-15 Combine like terms -10y+25 = 10y-15 sub 2y^2 from both sides -20y+25 = -15 sub 10y from each side -20y=-40 subtract 25 from each side y=2 divide each side by -20