Writing Equations of a

Line

Subtitle: What is the minimum information needed?

Various Forms of an Equation of a Line.

Slope-Intercept Form

Standard Form

Point-Slope Form

slope of the line

intercept

y mx b

m

b y

, , and are integers

0, must be postive

Ax By C

A B C

A A

1 1

1 1

slope of the line

, is any point

y y m x x

m

x y

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.

34

y = mx + b Use slope-intercept form.

y = x + (–2)34

Substitute for m and –2 for b.3

4

y = x (–2)34

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

y = x + 13

ANSWER

2. m = –2 , b = –4

y = –2x – 4

ANSWER

3. m = – , b =34

72

y = – x +34

72

ANSWER

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)

14

1m1

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

y – 3 = (x – (–2))14

Substitute for m2, x1, and y1.

y – 3 = (x +2)14 Simplify.

y – 3 = x +14

12

Distributive property

Write in slope-intercept form.

Write equations of parallel or perpendicular lines

1 7

4 2y x

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.

y = 4x + 10

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.

y = 3x – 14ANSWER

ANSWER

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