1. Algebra 1 Review
Casey Andreski
Bryce Lein

2. 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

3. Special cases
Cancel variables 3x+2=3(x-1) distribute 3x+2=3x-3 subtract 3x 2=-3 finished

4. Fractional Coefficient
1/2x /3x = 2 multiply by a common denominator
3x x = 12 add like terms
5x = divide by 5
X = finished

5. 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

6. Properties

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

8. 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

9. 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

10. Commutative Property of Addition
Example: a + b = b + a
Commutative Property of Multiplication
Example: a x b = b x a

11. Distributive Property (of Multiplication over Addition
Example: a x (b + c) = a x b + a x c

12. 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

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

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

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

16. 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

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

18. 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.

19. Product of Roots Property
Quotient of Roots Property

20. Root of a Power Property
Example:
Power of a Root Property
Example:

21. 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. =
Answer: Commutative Property (of Addition) 17 = 17

22. 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

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

24. 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

25. Disjunction 3x>18 or x<-1 X>6
3x>(14+4) or x<3-4 Click to see the answer
3x> or x<-1
X>6 -1 6

26. 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.

27. Slope Finding the slope with 2 given points Click for an example
m = Slope
Example:
(9,-3) (6,2)
Click for an example

28. Equations of Lines
Slope intercept form- Y = Mx + B Standard form – Ax + By = C Point slope form- Y – Y1 = M (X – X1)

29. 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

30. Graphing Lines
Slope intercept - y=-3x+7 7= y intercept -3 = slope

31. 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)

32. 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

33. 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

34. Addition/Subtraction Method (Elimination )
3x+5y=7
2x-4y=5
Multiply both equations to get either x or y to cancel
2(3x+5y)= = 6x+10y=14
3(2x-4y)= = x-12y= Subtract
22y= Divide by 22
y= -1/22

35. Terms
Dependent- both same line (Infinite solutions) Inconsistent- parallel lines (No solutions) Consistent- Intersecting lines (One solution)

36. 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.

37. 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

38. Factoring Trinomials GCF 2b+4b2+8b 2b(1+2b+4) Reverse foil x2+5x+6
PST x2-20x+25
(2x-5)2
Click for answers

39. 4 or More 3 by [(x1 x2+8x+16-3y2 (x+4)2-3y2 [(x+4)-3y] +4)-3y]
Click for answers
3 by [(x x2+8x+16-3y2
(x+4)2-3y2
[(x+4)-3y] +4)-3y]
2 by c3+bc+2c2+2b
c2(c+2)+b(c+2)
(c2+b) (c+2)

40. 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

41. Factor and Cancel =

42. Addition and subtraction of rational expressions
Click to see steps

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

44. 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

45. 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

46. Functions
Domain- Set of inputs Range- Set of outputs f(x)=2x-1 Domain – all real numbers Range – all real numbers

47. Functions
(1,1) and (0,-1) Are two ordered pairs of the linear function f(x)=2x-1

48. 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

49. 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.

50. Exponents
Property #1 x0 = 1 Example: 40 = 1 and ( )0 = 1 Property #2 xn × xm = xn + m Example: 46 × 45 = = 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

51. In the next slides you will review: Simplifying expressions with radicals – try to use all the root powers and don't forget rationalizing denominators

52. Expressions with Radicals

53. 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.)

54. 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-r r r2+15r-2700=0→(r-45)(r+60)=0 r=45 your rate was 45 mph

55. 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.

56. 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

57. 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 $ What was the price of the phone

58. 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.

59. 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