1. 2.4 Writing Equations for Linear Lines

2. 1. 2. Lesson 2.4, For use with pages 98-104
In a computer generated drawing, a line is represented by the equation 2x – 5y = 15. 1.
Solve the equation for y and identify the slope of the line. 5
ANSWER
y = 2
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?
ANSWER 5 – 2

3. EXAMPLE 1
Write an equation given the slope and y-intercept
Write an equation of the line shown.

4. Use slope-intercept form.
EXAMPLE 1
Write an equation given the slope and y-intercept
SOLUTION
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. 3 4 3 4
y = x (– 2)
Simplify.

5. Use slope - intercept form.
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 + 1 3
Substitute 3 for m and 1 for b.
y = x + 1 3
Simplify.

6. Use slope - intercept form.
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.
y = – 2x – 4
Simplify.

7. Use slope - intercept form.
GUIDED PRACTICE
for Example 1
3. m = – b = 3 4 7 2
SOLUTION
Use slope – intercept point form to write an equation of the line
y = mx + b
Use slope - intercept form.
y = – x + 3 4 7 2 3 4 7 2
Substitute – for m and for b.
y = – x + 3 4 7 2
Simplify.

8. Distributive property
EXAMPLE 2
Write an equation given the slope and a point
Write an equation of the line that passes through (5, 4) and has a slope of – 3.
SOLUTION
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
y = – 3x + 19
Write in slope-intercept form.

9. EXAMPLE 3
Write equations of parallel or perpendicular lines
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.

10. Distributive property
EXAMPLE 3
Write equations of parallel or perpendicular lines
y – y1 = m2(x – x1)
Use point-slope form.
y – 3 = – 4(x – (– 2))
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.

11. Distributive property
EXAMPLE 3
Write equations of parallel or perpendicular lines
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 m1
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 2
Distributive property
y = x + 1 4 2
Write in slope-intercept form.

12. Distributive property
GUIDED PRACTICE
GUIDED PRACTICE
for Examples 2 and 3
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 – y1 = m(x – x1)
Use point-slope form.
y – 6 = 4(x – (– 1))
Substitute for m, x1, and y1.
y – 6 = 4x + 4
Distributive property
y = 4x + 10
Write in slope-intercept form.

13. Distributive property Write in slope-intercept form.
GUIDED PRACTICE
GUIDED PRACTICE
for Examples 2 and 3
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 – y1 = m2(x – x1)
Use point-slope form.
y – (– 2) = 3(x – 4)
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.

14. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 2 and 3
b. A line perpendicular to a line with slope m1 = 3 has a slope of m2 = – = – 1 1 3 m1
Use point - slope form with (x1, y1) = (4, – 2)
y – y1 = m2(x – x1)
Use point-slope form.
y – (– 2) = – ( x – 4) 1 3
Substitute for m2, x1, and y1.
y + 2 = – ( x – 4) 1 3
Simplify. 4 3
y + 2 = – x – 1
Distributive property
y = – x – 1 3 2
Write in slope-intercept form.

15. EXAMPLE 4
Write an equation given two points
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 – y1
m =
x2 –x1
10 – (– 2) =
2 – 5 12
– 3
= – 4

16. Distributive property
EXAMPLE 4
Write an equation given two points
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
y = – 4x + 8
Write in slope-intercept form.

17. EXAMPLE 5
Write a model using slope-intercept form
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.

18. EXAMPLE 5
Write a model using slope-intercept form
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 The rate of change is the slope m.

19. Use (x1, y1) = (0, 2.00) and (x2, y2) = (10, 2.86). EXAMPLE 5
Write a model using slope-intercept form
y2 – y1
m =
x2 –x1
2.86 – 2.00 =
10 – 0
Use (x1, y1) = (0, 2.00)
and (x2, y2) = (10, 2.86).
0.86 = 10
= 0.086
STEP 3
Write a verbal model. Then write a linear equation.

20. EXAMPLE 5
Write a model using slope-intercept form
y = x
ANSWER
In slope-intercept form, a linear model is
y = 0.086x

21. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
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 – y1
m =
x2 –x1
– 7 – 5 =
4 – (– 2)
= – 2

22. Distributive property
GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
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.
Simplify
y – 7 = – 2 (x + 4)
y + 7 = – 2x + 8
Distributive property
y = – 2x + 1
Write in slope-intercept form.

23. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
7. (6, 1), (–3, –8)
SOLUTION
The line passes through (x1, y1) = (6, 1) and
(x2, y2) = (–3, –8). Find its slope.
y2 – y1
m =
x2 –x1
– 8 – 1 =
– 3 – 6
– 9
= 1

24. Distributive property
GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
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) = (– 3, – 8).
y – y1 = m(x – x1)
Use point-slope form.
y – (– 8)) = 1(x – (– 3))
Substitute for m, x, and y1.
y + 8 = 1 (x + 3)
Simplify
y + 8 = x + 3
Distributive property
y = x – 5
Write in slope-intercept form.

25. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
8. (–1, 2), (10, 0)
SOLUTION
The line passes through (x1, y1) = (– 1, 2) and
(x2, y2) = (10, 0). Find its slope.
y2 – y1
m =
x2 –x1
0 – 2 =
10– (– 1) 2 11 –

26. Distributive property
GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
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) = (10, 0).
y – y1 = m(x – x1)
Use point-slope form.
y – 0 = (x – 10 ) 2 11 –
Substitute for m, x, and y1.
y = (x – 10) 2 11 –
Simplify
y = 2 11 –
x + 20
Distributive property 2 11 –
x + 20 =
Write in slope-intercept form.

27. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
9. Sports In Example 5, the corresponding data for males are 3.42 million participants in 1993 and 3.99 million participants in Write a linear equation that models male participation in U.S. high school sports.

28. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 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 The rate of change is the slope m.

29. Use (x1, y1) = 3.42 and (x2, y2) = 3.99 GUIDED PRACTICE
for Examples 4 and 5
y2 – y1
m =
x2 –x1
3.99 – 3.42 =
10 – 0
Use (x1, y1) = 3.42
and (x2, y2) = 3.99 =
STEP 3
Write a verbal model. Then write a linear equation.

30. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
y = x
ANSWER
In slope-intercept form, a linear model is
y = 0.057x

31. EXAMPLE 6
Write a model using standard form
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.

32. EXAMPLE 6
Write a model using standard form
x y =
ANSWER
An equation for this situation is 0.79x y = 30.

33. GUIDED PRACTICE
for Example 6
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.

34. GUIDED PRACTICE
for Example 6
SOLUTION
x y =
ANSWER
An equation for this situation is .69x + .89y = 30.