§ 2.5

The Point-Slope Form of the Equation of a Line

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.5

Point-Slope Form

Point-Slope Form of the Equation of a Line

The point-slope equation of a nonvertical line with slope m that passes through the point is 11, yx

11 xxmyy

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.5

Point-Slope Form

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Write the point-slope form and then the slope-intercept form of the equation of the line with slope -3 that passes through the point (2,-4).

Substitute the given values 11 xxmyy

234 xy

634 xy23 xy

Distribute

Subtract 4 from both sides

234 xy Simplify

This is the equation of the line in point-slope form.

This is the equation of the line in slope-intercept form.

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.5

Point-Slope Form

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

First I must find the slope of the line. That is done as follows:

2

5

10

23

46

m

Write the point-slope form and then the slope-intercept form of the equation of the line that passes through the points (2,-4) and (-3,6).

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.5

Point-Slope Form

Now I can find the two forms of the equation of the line. In find the point-slope form of the line, I can use either point provided. I’ll use (2,-4).

Substitute the given values 11 xxmyy

224 xy

424 xyxy 3

Distribute

Subtract 4 from both sides

224 xy Simplify

This is the equation of the line in point-slope form.

This is the equation of the line in slope-intercept form.

CONTINUECONTINUEDD

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.5

Equations of Lines

11 xxmyy

Equations of LinesStandard Form Ax + By = C

Slope-Intercept Form y = mx + b

Horizontal Line y = b

Vertical Line x = a

Point-slope Form

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.5

Deciding which form to use:

Begin with the slope-intercept form if you know:

Begin with the point-slope form if you know:

The slope of the line and the y-intercept

or

Two points on the line, one of which is the y -intercept

The slope of the line and a point on the line other than the y-intercept

or

Two points on the line, neither of which is the y-intercept

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.5

Point-Slope Form

EXAMPLEEXAMPLE

Drugs are an increasingly common way to treat depressed and hyperactive kids. The line graphs represent models that show users per 1000 U.S. children, ages 9 through 17.

Y E A R

Use

rs P

er 1

000

Ch

ild

ren

1996

1997

1998

1999

2001

2000

5

2520

30

35

1510

(1995,24)

(1995,8)

1995

(2001,16.4)

(2001,34.2)

Antidepressants like Prozac

Stimulants like Ritalin

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.5

Point-Slope Form

(a) Find the slope of the blue line segment for children using stimulants. Describe what this means in terms of rate of change.

CONTINUECONTINUEDD

(b) Find the slope of the red line segment for children using antidepressants. Describe what this means in terms of rate of change.

(c) Do the blue and red line segments lie on parallel lines? What does this mean in terms of the rate of change for children using stimulants and children using antidepressants?

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.5

Point-Slope Form

CONTINUECONTINUEDD

This means that every year (since 1995), approximately 1.7 more children (per 1000) use stimulants like Ritalin.

7.16

2.10

19952001

242.34

m

SOLUTIONSOLUTION

(a) Find the slope of the blue line segment for children using stimulants. Describe what this means in terms of rate of change.

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.5

Point-Slope Form

CONTINUECONTINUEDD

This means that every year (since 1995), approximately 1.4 more children (per 1000) use antidepressants like Prozac.

4.16

4.8

19952001

84.16

m

(b) Find the slope of the red line segment for children using antidepressants. Describe what this means in terms of rate of change.

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.5

Point-Slope Form

CONTINUECONTINUEDD

The blue and red lines do not lie on parallel lines. This means that the number of child users for stimulants like Ritalin is increasing faster than the number of child users for antidepressants like Prozac.

(c) Do the blue and red line segments lie on parallel lines? What does this mean in terms of the rate of change for children using stimulants and children using antidepressants?

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.5

Parallel and Perpendicular Lines

Slope and Parallel Lines1) If two nonvertical lines are parallel, then they have the same slope.

2) If two distinct nonvertical lines have the same slope, then they are parallel.

3) Two distinct vertical lines, both with undefined slopes, are parallel.

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.5

Parallel and Perpendicular Lines

Slope and Perpendicular Lines1) If two nonvertical lines are perpendicular, then the product of their slopes is -1.

2) If the product of the slopes of two lines is -1, then the lines are perpendicular.

3) A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.5

Parallel and Perpendicular Lines

One line is perpendicular to another line if its slope is the negative reciprocal of the slope of the other line.

The following lines are perpendicular:

y = 2x + 6 and y = -(1/2)x – 4 are perpendicular.

y = -4x +5 and y = (1/4)x + 3 are perpendicular.

Blitzer, Intermediate Algebra, 5e – Slide #16 Section 2.5

Parallel and Perpendicular Lines

Two lines are parallel if they have the same slope.

The following lines are parallel:

y = 2x + 6 and y = 2x – 4 are parallel.

y = -4x +5 and y = -4x + 3 are parallel.

Blitzer, Intermediate Algebra, 5e – Slide #17 Section 2.5

Parallel and Perpendicular Lines

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Write an equation of the line passing through (2,-4) and parallel to the line whose equation is y = -3x + 5.

Since the line I want to represent is parallel to the given line, they have the same slope. Therefore the slope of the new line is also m = -3. Therefore, the equation of the new line is:

y – 2 = -3(x – (-4))

y – 2 = -3(x + 4)

y – 2 = -3x - 12

y = -3x - 10

Substitute the given values

Simplify

Distribute

Add 2 to both sides

Blitzer, Intermediate Algebra, 5e – Slide #18 Section 2.5

Parallel and Perpendicular Lines

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Write an equation of the line passing through (2,-4) and perpendicular to the line whose equation is y = -3x + 5.

The slope of the given equation is m = -3. Therefore, the slope of the new line is , since . Therefore, the using the slope m = and the point (2,-4), the equation of the line is as follows:

31 13 3

1

31

Blitzer, Intermediate Algebra, 5e – Slide #19 Section 2.5

Parallel and Perpendicular Lines

11 xxmyy

23

14 xy

23

14 xy

3

2

3

14 xy

43

2

3

1 xy

3

3

1

4

3

2

3

1 xy

3

12

3

2

3

1 xy

3

14

3

1 xy

CONTINUECONTINUEDD

Substitute the given values

Simplify

Simplify

Distribute

Subtract 4 from both sides

Common Denominators

Common Denominators