5.5

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Write Equations of Parallel and Perpendicular Lines

5.5 Warm-Up

Are the lines parallel? Explain.

2. –x = y + 4, 3x + 3y = 5

ANSWER

ANSWER

1. y – 2 = 2x, 2x + y = 7

Yes; both slopes are –1.

No; one slope is 2 and the other is –2.

5.5 Warm-Up

ANSWER $6

3. You play tennis at two clubs. The total cost C (in dollars) to play for time t (in hours) and rent equipment is given by C = 15t + 23 at one club andC = 15t + 17 at the other. What is the difference in total cost after 4 hours of play?

5.5 Example 1

SOLUTION

Write an equation of the line that passes through (–3, –5) and is parallel to the line y = 3x – 1.

STEP 1

Identify the slope. The graph of the given equation has a slope of 3. So, the parallel line through (–3, –5) has a slope of 3.

5.5 Example 1

STEP 2Find the y-intercept. Use the slope and the given point.

y = mx + b

–5 = 3(–3) + b

4 = b

Write slope-intercept form.

Substitute 3 for m, 3 for x, and 5 for y.

Solve for b.

STEP 3

Write an equation. Use y = mx + b.

y = 3x + 4 Substitute 3 for m and 4 for b.

5.5 Guided Practice

1. Write an equation of the line that passes through

(–2, 11) and is parallel to the line y = –x + 5.

y = –x + 9ANSWER

5.5 Example 2

Determine which lines, if any, are parallel or perpendicular.Line a: y = 5x – 3

Line b: x + 5y = 2

Line c: –10y – 2x = 0

SOLUTION

Find the slopes of the lines.

Line a: The equation is in slope-intercept form. The slope is 5.

Write the equations for lines b and c in slope-intercept form.

5.5 Example 2

Line b: x + 5y = 2

5y = – x + 2

Line c: –10y – 2x = 0

–10y = 2x

y = – x15xy = 2

515 +–

ANSWER

Lines b and c have slopes of – , so they are

parallel. Line a has a slope of 5, the negative reciprocal

of – , so it is perpendicular to lines b and c.

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5.5 Guided Practice

Determine which lines, if any, are parallel or perpendicular.Line a: 2x + 6y = –3

Line b: y = 3x – 8

Line c: –1.5y + 4.5x = 6

ANSWER

parallel: b and c; perpendicular: a and b, a and c

5.5 Example 3

SOLUTION

Line a: 12y = –7x + 42

Line b: 11y = 16x – 52

Find the slopes of the lines. Write the equations in slope-intercept form.

The Arizona state flag is shown in a coordinate plane. Lines a and b appear to be perpendicular. Are they?

STATE FLAG

5.5 Example 3

Line a: 12y = –7x + 42

Line b: 11y = 16x – 52

y = – x + 1242 7

12

1152

y = x –1611

ANSWER

The slope of line a is – . The slope of line b is .

The two slopes are not negative reciprocals, so lines a and b are not perpendicular.

712

1611

5.5 Guided Practice

3. Is line a perpendicular to line b? Justify your answer using slopes.

Line a: 2y + x = –12

Line b: 2y = 3x – 8

ANSWER

No; the slope of line a is – , the slope of line b is . The slopes are not negative reciprocals so the lines are not perpendicular.

12

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5.5 Example 4

SOLUTION

Write an equation of the line that passes through (4, –5) and is perpendicular to the line y = 2x + 3.

STEP 1

Identify the slope. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is .1

2–

5.5 Example 4

STEP 2 Find the y-intercept. Use the slope and thegiven point.

Write slope-intercept form.

–5 = – (4) + b12

Substitute – for m, 4 for x, and

–5 for y.

12

y = mx + b

–3 = b Solve for b.

STEP 3 Write an equation.

y = mx + b Write slope-intercept form.

y = – x – 312 Substitute – for m and –3 for b.1

2

5.5 Guided Practice

4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4x – 7.

y = – x + 414ANSWER

5.5 Lesson Quiz

1. Write an equation of the line that passes through the point (–1, 4) and is parallel to the line y = 5x – 2.

y = 5x + 9

ANSWER

Write an equation of the line that passes through the point (–1, –1) and is perpendicular to the line y = x + 2.1

4–

2.

y = 4x + 3

ANSWER

5.5 Lesson Quiz

3. Path a, b and c are shown in the coordinate grid. Determine which paths, if any, are parallel or perpendicular. Justify your answer using slopes.

ANSWER

Paths a and b are perpendicular because their slopes, 2 and are negative reciprocals. No paths are parallel.1

2–