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) Example: a + c = b + c Multiplication Property (of Equality) Example: If  a = b  then  a x c = b x c.

Reflexive Property (of Equality) Example: a = a Symmetric Property (of Equality) Example: a = b then b = a Transitive Property (of Equality) Example: If a = b and b = c, then a = c

Associative Property of Addition Example: a + (b + c) = (a + b) + c Associative Property of Multiplication Example: a x (b x c) = (a x b) x c

Commutative Property of Addition Example: a + b = b + a Commutative Property of Multiplication 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 Example: a + (-a) = 0 Prop of Reciprocals or Inverse Prop. of Multiplication Example: (b)1/b=1

Identity Property of Addition Example: y + 0 = y Identity Property of Multiplication Example: b x 1= b

Multiplicative Property of Zero Example: a x 0 = 0 Closure Property of Addition Example: 2 + 5 = 7 Closure Property of Multiplication Example: 4 x 5 = 20

Product of Powers Property Example: 42 x 44 = 46 Power of a Product Property Example: (2b)3 = 23 x b3 = 8b3

Quotient of Powers Property Example: 54/53 = 625/125 or 54-3 = 51 = 5 Power of a Quotient Property Example: (4/2)2 = 42/22 = 4

Negative Power Property Zero Power Property Example: a0 = 1 Negative Power Property Example: a-6 = 1/a6

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

Product of Roots Property Quotient of Roots Property

Root of a Power Property Example: Power of a Root Property Example:

Now you will take a quiz! Look at the sample problem and give the name of the property illustrated. Click when you’re ready to see the answer. 1. 14 + 3 = 3 + 14 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 sign B. Conjunction C. Disjunction

With only one inequality sign 3 + x < 3 + 2 Click when ready to see the answerer X < 2 2

Click when you’re ready to see the answer. Conjunction 3+5<1+x>-2-1 Click when you’re ready to see the answer. 8<1+x>-2-1 7<x>-4 -4 7

Disjunction 3x>18 or x<-1 X>6 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 variables In the next slides you will review: Linear equations in two variables Lots 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 Click for an example 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

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 Systems. A In the next slides you will review: Linear Systems A. Substitution Method B. 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 PST 4x2-20x+25 (2x-5)2 Click for answers

4 or More 3 by [(x1 x2+8x+16-3y2 (x+4)2-3y2 [(x+4)-3y] +4)-3y] 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 =

Addition and subtraction of rational expressions Click to see steps

Multiplication and division of rational expressions Click to see answer Division is multiplication of the reciprocal

In the next slides you will review: Functions. A. What does f(x) mean In the next slides you will review: Functions A. 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

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 Property #1 x0 = 1 Example: 40 = 1 and (2500000000000000000000)0 = 1 Property #2 xn × xm = xn + m Example: 46 × 45 = 46 + 5 = 411 Property #3 xn ÷ xm = xn − m Example: 46 ÷ 45 = 46 − 5 = 41 Property #4 (xn)m = xn × m (52)4 = 52 × 4 = 58 Property #5 (x × y)n = xn × yn (6 × 7)5 = 65 × 7 Property #6 x-n = 1 ÷(xn) = 1/(xn) 8-4 = 1 ÷ (84) = 1 / (84) Property #7 (x/y)n = xn / yn (8/5)4 = 84 / 54 Property #8 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

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.

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 Problem After 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

In the next slides you will review: Line of Best Fit or Regression Line A. 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