Warm-UpWarm-Up1. Graph the line y = 3x + 4.

5 minutes

2. Graph the line y = 3x - 23. What is the slope of the lines in the equations above?

Parallel and Parallel and Perpendicular LinesPerpendicular Lines

Objectives: •To determine whether the graphs of two equations are parallel•To determine whether the graphs of two equations are perpendicular

Parallel LinesParallel LinesParallel lines are lines in the same plane that never intersect.

-8 -6 -4 -2

2

42 6 8

4

6

-4

-6

-8

-2

8

Parallel lines have the same slope.

Example 1Example 1Determine whether these lines are parallel.y = 4x -6

and y = 4x + 2

The slope of both lines is 4.So, the lines are parallel.

Example 2Example 2Determine whether these lines are parallel.y – 2 = 5x + 4

and -15x + 3y = 9+2 +2

y = 5x + 6+15x +15x 3y = 9 +

15x3 3y = 3 + 5xy = 5x + 3

The lines have the same slope.So they are parallel.

Example 3Example 3Determine whether these lines are parallel.y = -4x + 2 and -5 = -2y + 8x

+2y + 2y2y - 5 = 8x

+5 +52y = 8x + 52 2

5y 4x 2

Since these lines have different slopes, they are not parallel.

PracticePractice

2) 3x – y = -5 and 5y – 15x = 10

Determine whether the graphs are parallel lines.

3) 4y = -12x + 16 and y = 3x + 4

1) y = -5x – 8 and y = 5x + 2

Example 4Example 4Write the slope-intercept form of the equation of the line passing through the point (1, –6) and parallel to the line y = -5x + 3.

slope of new line =

-5y – y1 = m(x – x1)

y – (-6) = -5(x – 1)y + 6 = -5x +

5 y = -5x - 1

PracticePracticeWrite the slope-intercept form of the equation of the line passing through the point (0,2) and parallel to the line 3y – x = 0.

Practice 2Practice 2Determine whether the graphs of the equations are parallel lines.

1) 3x – 4 = y and y – 3x = 8

2) y = -4x + 2 and -5 = -2y + 8x

Perpendicular LinesPerpendicular LinesPerpendicular lines are lines that intersect to form a 900 angle.

-8 -6 -4 -2

2

42 6 8

4

6

-4

-6

-8

-2

8

The product of the slopes of perpendicular lines is -1.

4m 22

2 1m 4 2

Example 1Example 1Determine whether these lines are perpendicular.

and y = -3x - 21y x 73

1m 3 m = -31 33 1

Since the product of the slopes is -1, the lines are perpendicular.

Example 2Example 2Determine whether these lines are perpendicular.

and y = -5x - 2

m 5 m = -55 5 25

Since the product of the slopes is not -1, the lines are not perpendicular.

y = 5x + 7

PracticePracticeDetermine whether these lines are perpendicular.1) 2y – x = 2 and y = -2x + 4

2) 4y = 3x + 12 and -3x + 4y – 2 = 0

Example 3Example 3Write an equation for the line containing (-3,-5) and perpendicular to the line y = 2x + 1.First, we need the slope of the line y = 2x + 1.

m = 2Second, we need to find out the slope of the line that is perpendicular to y = 2x + 1. 1m 2

Lastly, we use the point-slope formula to find our equation.

1 1(y y ) m(x x ) 1(y 5) (x 3)2

1y 5 (x 3)2

PracticePracticeWrite an equation for the line containing the given point and perpendicular to the given line.1) (0,0); y = 2x + 4

2) (-1,-3); x + 2y = 8