Writing Equations of
a Line
Various Forms of an Equation of a Line.
Gradient-Intercept Form
Standard Form
Point-Gradient Form
, , and are integers
0, must be postive
Ax By C
A B C
A A
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 c = –2. Use slope-intercept form to write an equation of the line.
34
y = mx + c Use gradient-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 gradient 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