Algebra 1 Review

Casey Andreski

Bryce Lein

In the next slides you will review:

Solving 1st power equations in one variable

A. Don't forget special cases where variables cancel to get {all reals} or

B. Equations containing fractional coefficients

C. Equations with variables in the denominator – remember to throw out answers that cause division by zero

Special cases

Cancel variables

3x+2=3(x-1) distribute

3x+2=3x-3 subtract 3x

2=-3 finished

Fractional Coefficient

• 1/2x - 3 + 1/3x = 2 multiply by a common denominator

• 3x - 18 + 2x = 12 add like terms

• 5x = 40 divide by 5

• X = 8 finished

Variables in the denominator

• 5/x + 3/4 = 1/2 Multiply by a common denominator

• 5 + 3/4x = 1/2x group like terms

• 5 = -3/4x + 2/4x add like terms

• 5 = -1/4x multiply by common denominator

• -20 = x

Properties

Addition Property (of Equality)

Multiplication Property (of Equality)

Example: a + c = b + c

Example:

If  a = b  then  a x c = b x c.

Reflexive Property (of Equality)

Symmetric Property (of Equality)

Transitive Property (of Equality)

Example:

a = a

Example:

a = b then b = a

Example:

If a = b and b = c, then a = c

Associative Property of Addition

Associative Property of Multiplication

Example:

a + (b + c) = (a + b) + c

Example:

a x (b x c) = (a x b) x c

Commutative Property of Addition

Commutative Property of Multiplication

Example:

a + b = b + a

Example:

a x b = b x a

Distributive Property (of Multiplication over Addition

Example:

a x (b + c) = a x b + a x c

Prop of Opposites or Inverse Property of Addition

Prop of Reciprocals or Inverse Prop. of Multiplication

Example:

a + (-a) = 0

Example:

(b)1/b=1

Identity Property of Addition

Identity Property of Multiplication

Example:

y + 0 = y

Example: b x 1= b

Multiplicative Property of Zero

Closure Property of Addition

Closure Property of Multiplication

Example: a x 0 = 0

Example: 2 + 5 = 7

Example: 4 x 5 = 20

Product of Powers Property

Power of a Product Property

Example: 42 x 44 = 46

Example: (2b)3 = 23 x b3 = 8b3

Quotient of Powers Property

Power of a Quotient Property

Example: 54/53 = 625/125 or 54-3 = 51 = 5

Example: (4/2)2 = 42/22 = 4

Zero Power Property

Negative Power Property

Example: a0 = 1

Example: a-6 = 1/a6

Zero Product Property

Example: If ab = 0 , then either a = 0 or b = 0.

Product of Roots Property

Quotient of Roots Property

a b a b

a a

bb

Root of a Power Property

Power of a Root Property

Example:

Example:

2x x

2x

Now you will take a quiz!Look at the sample problem and give the name of the property illustrated.

1. 14 + 3 = 3 + 14

Click when you’re ready to see the answer.

Answer: Commutative Property (of Addition)

17 = 17

In the next slides you will review:

Solving 1st power inequalities in one variable. (Don't forget the special cases of {all reals} and )A. With only one inequality signB. ConjunctionC. Disjunction

With only one inequality sign

3 + x < 3 + 2

Click when ready to see the answerer

2

X < 2

Conjunction

3+5<1+x>-2-1Click when you’re ready to see the answer.

8<1+x>-2-1

7<x>-4

7-4

Disjunction 3x>(14+4) or x<3-4 Click to see the answer

3x>18 or x<-1

X>6

-1 6

In the next slides you will review:

Linear equations in two variablesLots to cover here: slopes of all

types of lines; equations of all types of lines, standard/general form, point-slope form, how to graph, how to find intercepts, how and when to use the point-slope formula, etc. Remember you can make lovely graphs in Geometer's Sketchpad and copy and paste them into PPT.

Slope

Finding the slope with 2 given points

m = Slope

Example:

(9,-3) (6,2)

2-9 -7

6+3 9

Click for an

example

Equations of Lines

Slope intercept form- Y = Mx + B

Standard form – Ax + By = C

Point slope form- Y – Y1 = M (X – X1)

Graphing Lines Point Slope- use this when you only have 2

points.

First : find the slope

Next put the equation into point slope form:

y-y1=m(x-x1)

Example: (3,5) (2,1)

Slope: = 4

Y-5=4(x-3) = y-5=4x-12 = y=4x-7

5 1

3 2

Graphing Lines

Slope intercept - y=-3x+7

7= y intercept

-3 = slope

Graphing Lines

Standard form - 3x + 2y = 6

Set x to zero to find y

Set y to zero to find x

Points : (2,0) (0,3)

In the next slides you will review:

Linear SystemsA. Substitution MethodB. Addition/Subtraction

Method (Elimination ) C. Check for understanding

of the terms dependent, inconsistent and consistent

Substitution Method 4x-5y=12

Y=2x-8

Put (2x-8) in for y for the top equation

Click for solution 4x-5(2x-8)=12 Distribute

4x-10x+40=12 add/subtract common terms

-6x=28 Divide

X= -3/14

Addition/Subtraction Method (Elimination )

3x+5y=7

2x-4y=5

Multiply both equations to get either x or y to cancel

2(3x+5y)=7 = 6x+10y=14

3(2x-4y)=5 = 6x-12y=15 Subtract

22y=-1 Divide by 22

y= -1/22

Terms

Dependent- both same line (Infinite solutions)

Inconsistent- parallel lines (No solutions)

Consistent- Intersecting lines (One solution)

In the next slides you will review:

Factoring – since we just completed the Inspiration Project on this topic, just summarize all the factoring methods quickly. Note that you will be using your factoring methods in areas 7 & 8 below so no need to include extra practice problems here.

Factoring Binomials

difference of squares 49x4-9y2

(7x2+3y) (7x2-3y) sum and diff of squares a3-27 (a-3) (a2+3a+9)

click for answers

Factoring Trinomials

GCF 2b+4b2+8b

2b(1+2b+4)

Reverse foil x2+5x+6

(x+3) (x+2)

PST 4x2-20x+25

(2x-5)2

Click for answers

4 or More Click for answers

3 by [(x1 x2+8x+16-3y2

(x+4)2-3y2

[(x+4)-3y] +4)-3y] 2 by 2 c3+bc+2c2+2b

c2(c+2)+b(c+2)

(c2+b) (c+2)

In the next slides you will review:

Rational expressions – try to use all your factoring methods somewhere in these practice problems

A. Simplify by factor and cancel

B. Addition and subtraction of rational expressions

C. Multiplication and division of rational expressions

Factor and Cancel

16

42

x

x4

1

x=

Addition and subtraction of rational expressions

416

22

x

x

x

x

16

)4(22

x

xxx16

422

2

x

xxx

16

62

2

x

xx

Click to see steps

Multiplication and division of rational expressions

6

3

6

442

2

x

x

xx

xx

6

2

6

3

)3)(2(

)2)(2(

x

x

x

x

xx

xx

Click to see answer

Division is multiplication of the reciprocal

In the next slides you will review:

FunctionsA. What does f(x) mean? Are all

relations function?B. Find the domain and range of a

function.C. Given two ordered pairs of data,

find a linear function that contains those points.

D. Quadratic functions – explain everything we know about how to graph a parabola

Functions

f(x) means that f is a function of x

All functions are relations but not all relations are functions

A function is 1 to 1 which means for each input there is exactly one output

Functions

Domain- Set of inputs

Range- Set of outputs

f(x)=2x-1

Domain – all real numbers

Range – all real numbers

Functions

(1,1) and (0,-1)

Are two ordered pairs of the

linear function f(x)=2x-1

Quadratic functions

f(x)=ax2+bx+c

Vertex x= , then solve for f(x)

X-intercepts set f(x) equal to zero factor and solve for x

y-intercepts Set x to zero and solve for f(x)

line of symmetry the line of

a

b

2

a

b

2

a

b

2

In the next slides you will review:

Simplifying expressions with exponents – try to use all the power properties and don't forget zero and negative powers.

Exponents

www.basic-mathematics.com

In the next slides you will review:

Simplifying expressions with radicals – try to use all the root powers and don't forget rationalizing denominators

Expressions with Radicals

2

2

22

21

2

1

2322282

2

1

4

1

8

2

41682

In the next slides you will review:

Minimum of four word problems of various types. You can mix these in among the topics above or put them all together in one section. (Think what types you expect to see on your final exam.)

Word Problem

You drove 180 miles at a constant rate and it took you t hours. If you would have driven 15 mph faster you would have saved an hour. What was your rate?

180 = rt → t = 180/r

180 = (r +15)(t –1)→180= (r+15)(180/r – 1)

180r = (r+15)(180 – r)→180r=180r-r2+2700-15r

r2+15r-2700=0→(r-45)(r+60)=0

r=45 your rate was 45 mph

Word Problem

If Joe can shovel his driveway in 2 hours and Bill can do it in 3 hours, how long will it take for both of them to shovel the driveway.

5

11

65

623

132

x

x

xx

xx

Word Problem

If 2 t-shirts and 3 pairs of shorts cost $69, and 2 pair of shorts are $30. How Much is a t-shirt?

2t+3s=69

2s=30

s=15

2t+3(15)=69

2t+45=69

2t=24

t=12

Word ProblemAfter bill lost his cell phone he had to pay his

parents 28% of the cost to buy a new phone. Bill had to pay $21.28. What was the price of the phone

76$

7628.

28.21

28.2128.

p

p

p

In the next slides you will review:

Line of Best Fit or Regression LineA. When do you use this?B. How does your calculator

help?C. Give a set of sample data in

question format to see if your students can find the regression equation.

Line of best fit or regression

You use to come up with a linear equation that best fits the data.

Put the input in list 1 and the out put in list 2

Then hit stat calc

Next hit 4:linreg(ax=b)

Y=ax+b is the line of best fit for the data

Question

What is the line of best fit for the given data points?

(0,5) (1,9) (-1,4) (-3,0) (-2,1) (3,13)

Y=1.5x+4.8