Linear Equations in Two Variables3.33.33.33.3

0

y bm

x

y bm

x

mx y b

mx b y

y mx b

Slope-Intercept Form

Slide 3.3- 2

Slope-Intercept Form

The slope-intercept form of the equation of a line with slope m and y-intercept (0, b) is

y = mx + b.

Slope y-intercept is (0, b).

Slide 3.3- 3

EXAMPLE 1

Find an equation of the line with slope 2 and

y-intercept (0, –3).

m = 2

b = –3

Substitute these values into the slope-intercept form.

y = mx + b

y = 2x – 3

Slide 3.3- 4

EXAMPLE 2

Graph the line, using the slope and y-intercept.

x + 2y = –4

Write the equation in slope-intercept form by solving for y.

x + 2y = –4 2y = –x – 4 Subtract x.

1

22y x

Slope y-intercept (0, –2)

Divide by 2

Slide 3.3- 5

continued

Graph:

1. Plot the y-intercept.

(0, -2)

2. The slope is

3. Using (-1/2), begin at (0,-2) and move 1 unit down and 2 units right.

4. The line through these two points is the required graph.

1

22y x

1 1 or .

2 2

Slide 3.3- 6

Point-Slope Form

The point-slope form of the equation of a line with slope m passing through the point (x1, y1) is

y – y1 = m(x – x1).

Slope

Given point

If you do not like to deal with fractions, you can use your slope formula as well.

Slide 3.3- 7

EXAMPLE 3

Find an equation of the line with slope 2/5 and passing through the point (3, –4).

Use the point-slope form with (x1, y1) = (3, –4) and m = 2/5. 1 1( )y xmy x

2

5( )4) ( 3y x

24 ( 3)

5y x

5 20 2 6y x

2 5 26x y 2 5 26x y

Substitute

Multiply by 5.

Subtract 2x and 20.

Multiply by -1.

Slide 3.3- 8

EXAMPLE 4

Find an equation of the line passing through the points (–2, 6) and (1, 4). Write the equation in standard form.

First find the slope by the slope formula.

Use either point as (x1, y1) in the point-slope form of the equation of a line.

Using the point (1, 4): x1 = 1 and y1 = 4

4 6 2

1 ( 2 3

2

3)m

Slide 3.3- 9

continued

1 1( )y xmy x

2

3(4 1)y x

3 12 2 2y x

2 3 14x y

Substitute

Multiply by 3.

Add 2x and 12.

m = -2/3; x1 = 1 and y1 = 4

If the other point were used, the same equation would result.

Slide 3.3- 10

Equations of Horizontal and Vertical LinesThe horizontal line through the point (a, b) has equation y = b.

The vertical line through the point (a, b) has equation

x = a.

Slide 3.3- 11

EXAMPLE 5

Find an equation of the line passing through the point (–8, 3) and

a. parallel to the line 2x – 3y = 10;

b. perpendicular to the line 2x – 3y = 10.

Write each equation in slope-intercept form.

a. Find the slope of the line 2x – 3y = 10 by solving for y. 2 3 10x y

3 2 10y x 1

3

2

3

0y x

Slide 3.3- 12

continued

The slope is

Parallel lines have the same slope. Use point slope form and the given point.

2

3

Find an equation of the line passing through the point (–8, 3).

1 1( )y xmy x

23 [ ( 8)]

3y x

23 ( 8)

3y x

2 163

3 3y x

2 16 9

3 3 3y x

2 25

3 3y x

The fractions were not cleared because we want the equation in slope-intercept form instead of standard form.

Slide 3.3- 13

continued

b. Perpendicular lines. The slope is the negative reciprocal of

Use point slope form and the given point. m =

2.

3

Find an equation of the line passing through the point (–8, 3).

1 1( )y xmy x

33 [ ( 8)]

2y x

33 ( 8)

2y x

33 12

2y x 3

92

y x

3

2

Slide 3.3- 14

Slide 3.3- 15

EXAMPLE 6

Suppose there is a flat rate of $0.20 plus a charge of $0.10 per minute to make a telephone call. Write an equation that gives the cost y for a call of x minutes.

y = $0.20 + $0.10x

or y = $0.10x + 0.20

Slide 3.3- 16

EXAMPLE 7The percentage of the U.S. population 25 years and older with at least a high school diploma is shown in the table for selected years. Find an equation that models the data, using x = 0 to represent 1940, x = 10 to represent 1950, and so on. Year Percent

1940 24.5

1950 34.3

1960 41.4

1970 52.3

1980 66.5

1990 75.2

2000 80.4

Slide 3.3- 17

continuedChoose two data points and find the slope. Use 1940 and 2000.

The y-intercept is (0, 24.5).

The equation is:

y = 0.93x + 24.5

Selecting two different ordered

pairs will lead to a different equation.

Year Percent

1940 24.5

1950 34.3

1960 41.4

1970 52.3

1980 66.5

1990 75.2

2000 80.4

80.4 24.5 55.90.9316

2000 1940 60m

Slide 3.3- 18

EXAMPLE 8

Use the ordered pairs (11, 164) and (13, 203) to find an equation that models the data in the graph below.

Use the point-slope form with (11, 164).

203 164 3919.5

13 11 2m

1 1( )y y m x x

164 19.5( 11)y x

164 19.5 214.5y x

19.5 50.5y x

Slide 3.3- 19

Homework

pg 186 # 4, 6-90 m6, 97, 98

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