–10 –705

20

–3–15

x y

Point-Slope Form and Writing Linear Equations

LESSON 5-4 Lesson Quiz

23

25yes; y + 3 = (x – 0)

65

65

75

y + 5 = – (x – 3); y = – x –

y – 4 = – (x – 0), or y = – x + 423

23

1. Graph the equation y + 1 = –(x – 3).

2. Write an equation of the line with

slope – that passes through

the point (0, 4).

3. Write an equation for the line that passes through (3, –5) and (–2, 1) in point-slope form and slope-intercept form.

4. Is the relationship shown by the data linear? If so, model that data with an equation.

5-5Parallel and

Perpendicular Lines

California Content Standards7.0 Derive linear equations by using the point-slope formula. Master8.0 Understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Find the equation of a line perpendicular to a given line that passes through a given point. Develop Master

Page 86

What You’ll Learn…And Why

What You’ll Learn…And Why

To determine whether lines are parallelTo determine whether lines are

perpendicular

To use parallel and perpendicular lines to plan a bike path.

Slopes of Parallel Lines

Nonvertical lines are parallel if

Any two vertical lines are parallel.

Example the equations

Have the same slope, , and different y-intercept. The graphs of the two equations are parallel.

They have the same slope and different y-intercepts

2

3

y =23x+1 and y=

23x−3

Slopes of Perpendicular Lines

Two lines are perpendicular if the product of their slopes is

A vertical and horizontal line are also perpendicular.

Example The slope of

The slope of y = 4x + 2 is 4. Since

The graphs of the two equations are perpendicular.

- 1

y =−14x−1 is −

14.

−1

4• 4 = −1

VocabularyParallel lines are lines in the same plane that

never intersect. Equations have the same slope, different y-intercepts.

Perpendicular lines are lines that intersect to form right angles. Equations have the opposite and reciprocal slopes.

The product of a number and its negative reciprocal is -1.

Write an equation for the line that contains (–2, 3) and is parallel to y = x – 4.

52

52

Step 1  Identify the slope of the given line.

y = x – 4

slope

Step 2 Write the equation of the line through (–2, 3) using slope-intercept form.

Write an equation for the line that contains

(–2, 3) and is parallel to y = x – 4.

y – 3 = x + 5 Simplify.52

y = x + 8 Add 3 to each side and simplify.52

y – 3 = (x + 2) Substitute (2,3) for (x1, y1) and

for m.

52

52

52

(Continued)

y – y1 = m(x – x1) Use point-slope form.

The line in the graph represents the street in front of a new house. The point is the front door. The sidewalk from the front door will be perpendicular to the street. Write an equation representing the sidewalk.

Step 1  Find the slope m of the street.

m = = = = – Points (0, 2) and (3, 0) are on the street.

y2 – y1

x2 – x1

0 – 23 – 0

–2 3

23

The equation for the sidewalk is y = x – 3.32

Step 2 Find the negative reciprocal of the slope.

The negative reciprocal of – is . So the slope of the

sidewalk is . The y-intercept is –3.

23

32

32

CA Standards Check1) Write an equation for the line that contains (2,-6) and

is parallel to y=3x +9.

Parallel lines: same slope, different y-intercepts.

PointSlope

y – y1 = m (x – x1 ) - 6 2 3

y + 6 = 3(x – 2 )

y + 6 = 3x – 6 – 6 – 6

y = 3x – 12

CA Standards Check2a) Write an equation for the line that contains (1,8)

and is perpendicular to y=3/4x + 1.

Perpendicular lines: Opposite and reciprecal slopes.

PointSlope

y – y1 = m (x – x1 ) 8 1 −4

3

y – 8 = (x – 1 ) −4

3

y – 8 = x + −4

34

3

y = x + −4

3

28

3

8 =3

24

+ 8 + .24

3

y =−43x+ 9

13

CA Standards Check2b) Using the diagram in Ex 2, rite an equation in slope-intercept

form for a new sidewalk perpendicular to the street from a front door at (-1, -2).

Point

Slope y – y1 = m (x – x1 ) (-1) (-2) −2

3

y + 1 = (x + 2 ) −2

3 -1 =3

-3

y = x - 3 3

2

y + 1 = x - −2

34

3

- 1 - . 3

3

y = x - −2

3

8

3 y =−23x−2

23

Closure:

Compare the equations of non-vertical parallel line, and perpendicular lines.

Parallel lines have the same slope, but different y-intercepts.Perpendicular lines have slopes that are negative reciprocals of each other.