2.4Writing Equations for

Linear Lines

Lesson 2.4, For use with pages 98-104

In a computer generated drawing, a line is represented by the equation 2x – 5y = 15.

ANSWER 5–2

1. Solve the equation for y and identify the slope of the line.

5ANSWER y =

2 25

x – 3;

2. What should the slope of a second line in the drawing be if that line must be perpendicular to the first line?

Write an equation given the slope and y-interceptEXAMPLE 1

Write an equation of the line shown.

SOLUTION

Write an equation given the slope and y-interceptEXAMPLE 1

From the graph, you can see that the slope is m = and the y-intercept is b = – 2. Use slope-intercept form to write an equation of the line.

3

4

y = mx + b Use slope-intercept form.

y = x + (– 2)3

4 Substitute for m and –2 for b.34

y = x (– 2)3

4 Simplify.

GUIDED PRACTICE for Example 1

Write an equation of the line that has the given slope and y - intercept.

1. m = 3, b = 1

SOLUTION

Use slope – intercept point form to write an equation of the line

y = mx + b Use slope - intercept form.

y = x + 13 Substitute 3 for m and 1 for b.

Simplify.y = x + 13

GUIDED PRACTICE for Example 1

2. m = – 2 , b = – 4

SOLUTION

Use slope – intercept point form to write an equation of the line

y = mx + b Use slope - intercept form.

y = – 2x + (– 4 ) Substitute – 2 for m and – 4 for b.

Simplify.y = – 2x – 4

GUIDED PRACTICE for Example 1

SOLUTION

Use slope – intercept point form to write an equation of the line

y = mx + b Use slope - intercept form.

3. m = – b =3

4

7

2

Substitute – for m and for b.34

Simplify.

y = – x +3

4

7

272

y = – x +3

4

7

2

Write an equation given the slope and a pointEXAMPLE 2

Write an equation of the line that passes through (5, 4) and has a slope of – 3.

Because you know the slope and a point on the line, use point-slope form to write an equation of the line. Let (x1, y1) = (5, 4) and m = – 3.

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

y – 4 = – 3(x – 5) Substitute for m, x1, and y1.

y – 4 = – 3x + 15 Distributive property

SOLUTION

y = – 3x + 19 Write in slope-intercept form.

EXAMPLE 3

Write an equation of the line that passes through (–2,3) and is (a) parallel to, and (b) perpendicular to, the line y = –4x + 1.

SOLUTION

a. The given line has a slope of m1 = –4. So, a line parallel to it has a slope of m2 = m1 = –4. You know the slope and a point on the line, so use the point-slope form with (x1, y1) = (– 2, 3) to write an equation of the line.

Write equations of parallel or perpendicular lines

EXAMPLE 3

y – 3 = – 4(x – (– 2))

y – y1 = m2(x – x1) Use point-slope form.

Substitute for m2, x1, and y1.

y – 3 = – 4(x + 2) Simplify.

y – 3 = – 4x – 8 Distributive property

y = – 4x – 5 Write in slope-intercept form.

Write equations of parallel or perpendicular lines

EXAMPLE 3

b. A line perpendicular to a line with slope m1 = – 4 has a slope of m2 = – = . Use point-slope form with

(x1, y1) = (– 2, 3)

1

4

1m1

y – y1 = m2(x – x1) Use point-slope form.

y – 3 = ( x – (– 2))1

4 Substitute for m2, x1, and y1.

y – 3 = ( x +2)1

4 Simplify.

y – 3 = x +1

4

1

2 Distributive property

Write in slope-intercept form.y = x +1

4

1

2

Write equations of parallel or perpendicular lines

GUIDED PRACTICE for Examples 2 and 3GUIDED PRACTICE

4. Write an equation of the line that passes through (– 1, 6) and has a slope of 4.

SOLUTION

Because you know the slope and a point on the line, use the point-slope form to write an equation of the line. Let (x1, y1) = (–1, 6) and m = 4

y – 6 = 4(x – (– 1))

y – y1 = m(x – x1) Use point-slope form.Substitute for m, x1, and y1.

y – 6 = 4x + 4 Distributive property

y = 4x + 10 Write in slope-intercept form.

GUIDED PRACTICE for Examples 2 and 3GUIDED PRACTICE

5. Write an equation of the line that passes through (4, –2) and is (a) parallel to, and (b) perpendicular to, the line y = 3x – 1.

SOLUTION

The given line has a slope of m1 = 3. So, a line parallel to it has a slope of m2 = m1 = 3. You know the slope and a point on the line, so use the point - slope form with (x1, y1) = (4, – 2) to write an equation of the line.

y – (– 2) = 3(x – 4)

y – y1 = m2(x – x1) Use point-slope form.

Substitute for m2, x1, and y1.

y + 2 = (x – 4) Simplify.y + 2 = 3x – 12 Distributive property

y = 3x – 14 Write in slope-intercept form.

GUIDED PRACTICE for Examples 2 and 3GUIDED PRACTICE

y – y1 = m2(x – x1) Use point-slope form.

Substitute for m2, x1, and y1.

Simplify.

Distributive property

Write in slope-intercept form.

Use point - slope form with (x1, y1) = (4, – 2)

y – (– 2) = – ( x – 4)1

3

y + 2 = – ( x – 4)1

3

y = – x – 1

3

2

3

1

3

b. A line perpendicular to a line with slope m1 = 3 has a slope of m2 = – = –

1

m1

4

3y + 2 = – x –

1

3

Write an equation given two points

EXAMPLE 4

Write an equation of the line that passes through (5, –2)and (2, 10).

SOLUTION

The line passes through (x1, y1) = (5,– 2) and (x2, y2) = (2, 10). Find its slope.

y2 – y1m =

x2 –x1

10 – (– 2) =

2 – 5

12– 3

= = – 4

Write an equation given two points

EXAMPLE 4

You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line. Choose (x1, y1) = (4, – 7).

y2 – y1 = m(x – x1) Use point-slope form.

y – 10 = – 4(x – 2) Substitute for m, x1, and y1.

y – 10 = – 4x + 8 Distributive property

Write in slope-intercept form.y = – 4x + 8

Write a model using slope-intercept formEXAMPLE 5

Sports

In the school year ending in 1993, 2.00 million females participated in U.S. high school sports. By 2003,the number had increased to 2.86 million. Write a linear equation that models female sports participation.

Write a model using slope-intercept formEXAMPLE 5

SOLUTION

STEP 1

Define the variables. Let x represent the time (in years) since 1993 and let y represent the number of participants (in millions).

STEP 2

Identify the initial value and rate of change. The initial value is 2.00. The rate of change is the slope m.

Write a model using slope-intercept formEXAMPLE 5

y2 – y1m =

x2 –x1

2.86 – 2.00 = 10 – 0

0.86 = 10

= 0.086

Use (x1, y1) = (0, 2.00)

and (x2, y2) = (10, 2.86).

STEP 3

Write a verbal model. Then write a linear equation.

Write a model using slope-intercept formEXAMPLE 5

y = 2.00 + 0.086 x

ANSWER

In slope-intercept form, a linear model is y = 0.086x + 2.00.

GUIDED PRACTICE for Examples 4 and 5GUIDED PRACTICE

Write an equation of the line that passes through the given points.

6. (– 2, 5), (4, – 7)

SOLUTION

The line passes through (x1, y1) = (– 2, 5) and (x2, y2) = (4, – 7). Find its slope.

y2 – y1m =

x2 –x1

– 7 – 5 =

4 – (– 2)= – 2

GUIDED PRACTICE for Examples 4 and 5GUIDED PRACTICE

You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line. Choose (x1, y1) = (4, – 7).

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

y – 7 = – 2(x – 4) Substitute for m, x1, and y1.

y – 7 = – 2 (x + 4)

Distributive property

Write in slope-intercept form.y = – 2x + 1

y + 7 = – 2x + 8

Simplify

GUIDED PRACTICE for Examples 4 and 5GUIDED PRACTICE

7. (6, 1), (–3, –8)

SOLUTION

The line passes through (x1, y1) = (6, 1) and (x2, y2) = (–3, –8). Find its slope.

y2 – y1m =

x2 –x1

– 8 – 1 =

– 3 – 6

– 9– 9

= = 1

GUIDED PRACTICE for Examples 4 and 5GUIDED PRACTICE

You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line.

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

Substitute for m, x, and y1.

Distributive property

Write in slope-intercept form.y = x – 5

Choose (x1, y1) = (– 3, – 8).

Simplify

y – (– 8)) = 1(x – (– 3))

y + 8 = 1 (x + 3)

y + 8 = x + 3

GUIDED PRACTICE for Examples 4 and 5GUIDED PRACTICE

8. (–1, 2), (10, 0)

SOLUTION

The line passes through (x1, y1) = (– 1, 2) and (x2, y2) = (10, 0). Find its slope.

y2 – y1m =

x2 –x1

0 – 2 =

10– (– 1)

211

= –

GUIDED PRACTICE for Examples 4 and 5GUIDED PRACTICE

You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line.

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

Substitute for m, x, and y1.

Distributive property

Write in slope-intercept form.

Choose (x1, y1) = (10, 0).

Simplify

y – 0 = (x – 10 ) 211

–

y = (x – 10)211

–

y = 2

11– x +

2011

211

– x +2011=

GUIDED PRACTICE for Examples 4 and 5GUIDED PRACTICE

9. Sports In Example 5, the corresponding data for males are 3.42 million participants in 1993 and 3.99 million participants in 2003. Write a linear equation that models male participation in U.S. high school sports.

GUIDED PRACTICE for Examples 4 and 5GUIDED PRACTICE

SOLUTION

STEP 1

Define the variables. Let x represent the time (in years) since 1993 and let y represent the number of participants (in millions).

STEP 2

Identify the initial value and rate of change. The initial value is 3.42. The rate of change is the slope m.

GUIDED PRACTICE for Examples 4 and 5GUIDED PRACTICE

y2 – y1m =

x2 –x1

3.99 – 3.42 = 10 – 0

= 0.057

Use (x1, y1) = 3.42

and (x2, y2) = 3.99

STEP 3

Write a verbal model. Then write a linear equation.

GUIDED PRACTICE for Examples 4 and 5GUIDED PRACTICE

y = 3.42 + 0.057 x

ANSWER

In slope-intercept form, a linear model is y = 0.057x + 3.42

Write a model using standard form

EXAMPLE 6

Online Music

You have $30 to spend on downloading songs for your digital music player. Company A charges $.79 per song, and company B charges $.99 per song. Write an equation that models this situation.

SOLUTION

Write a verbal model. Then write an equation.

Write a model using standard formEXAMPLE 6

0.79 x + 0.99 y = 30

ANSWER

An equation for this situation is 0.79x + 0.99y = 30.

GUIDED PRACTICE for Example 6

10. What If? In Example 6, suppose that company A charges $.69 per song and company B charges $.89 per song. Write an equation that models this situation.

GUIDED PRACTICE for Example 6

.69 x + .89 y = 30

ANSWER

An equation for this situation is .69x + .89y = 30.

SOLUTION