CRCT Review1. Three runners are in a race. If they all finish at different times, in

how many ways can they finish the race in first, second and third place?a. 3 b. 6 c. 9 d. 12

2. Which of the following is the set of integers greater than -5 and less than 2?a. {-5, -4, -3, -2, -1, 0, 1, 2} b. {-5, -4, -3, -2, -2, -1, 1, 2}c. {-4, -3, -2, -1, 0, 1} d. {-4, -3, -2, -1, 1}

3. If Nora rolls a number cube with faces numbered 1 through 6 and spins a spinner with 8 equal sections labeled A through H, what is the probability that the number cube will show a number greater than 1 and the spinner will land on a vowel?a. b. c. d.

1

2

5

24

5

12

1

3

Parallel Lines

• Two lines with the same slope are said to be parallel lines. If you graph them they will never intersect.

• We can decide algebraically if two lines are parallel by finding the slope of each line and seeing if the slopes are equal to each other.

• We can find the equation of a line parallel to a given line and going through a given point by:

• a) first finding the slope m of the given line; • b.) finding the equation of the line through the given

point with slope m.

Testing if Lines are Parallel

Are the lines parallel?

12 3 9 and -8 2 14x y x y

Find the slope of 12 3 9

3 12 9

4 3

x y

y x

y x

The slope m = -4

Find the slope of 8 2 14

2 8 14

4 7

x y

y x

y x

The slope m = -4

Since the slopes are equal the lines are parallel.

Practice Testing if Lines are Parallel

Are the lines 6 3 5 and 2 4 4x y y x parallel?

6 3 5

3 6 5

52 32

x y

y x

y x

m

2 4 4

2 2

2

y x

y x

m

Since the slopes are differentthe lines are not parallel.

Are the lines 2 4 and 2 4 12x y x y parallel?

2 4

2 4

1 221

2

x y

y x

y x

m

2 4 12

4 2 12

1 321

2

x y

y x

y x

m

Since the slopes are equalthe lines are parallel.

Parallel Equations• Parallel lines have the same slope• Given: y = 3x + 6, write an equation parallel that passes through (4,

7)m = 3.Use y = mx + b to find the other equation.Substitute m =3; x = 4; y = 7 to find b. 7 = 3(4) + b 7 = 12 + b-12 -12 -5 = bThe new equation with point (4,7) is:

y = 3x - 5

Testing if Lines Are Perpendicular

1Are the lines 2 5 and 4 perpendicular?

2x y y x

Find the slope of 2 5 2

2 5

x y m

y x

1 1Find the slope of 4

2 2y x m

Since the slopes are negative reciprocals of each other the lines are perpendicular. 1

2 12

Practice Testing if Lines Are Perpendicular

Are the lines 6 3 5 and 2 4 4 perpendicular?x y y x 6 3 5

3 6 5

52 32

x y

y x

y x

m

2 4 4

2 2

2

y x

y x

m

Since the slopes are not negative reciprocals of each other (their product is not -1) the lines are not perpendicular

Are the lines 2 4 and 4 2 6 perpendicular?x y x y 2 4

2 4

1 221

2

x y

y x

y x

m

4 2 6

2 4 6

2 3

2

x y

y x

y x

m

Since the slopes are negative reciprocals of each other (their product is -1) the lines are perpendicular.

Perpendicular Equations• Perpendicular lines have the “opposite reciprocal” slope.• Given: y = -3x + 1; write an equation perpendicular passing through (6, -1)So, m = 1/3. Use y = mx + b to find the other equation.Substitute m = 1/3; x = 6; y = -1 to find b. -1 = 1/3(6) + b -1 = 6/3 + b -1 = 2 + b -2 -2 -3= bThe new equation with point (6,-1) is:

y = 1/3x - 3