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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.
15
15
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
32
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–
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