Slope-Intercept Form for the Equation of a Line
We have found the equation of a line in point-slope form and in standard
form.
Another form for the equation of a line is the slope-intercept form. We
can derive this from the point-slope form.
| We begin with the point-slope form for the
equation of a line. Use the y-intercept, (0, b), for the point (x1, y1).That is, substitute 0 for
x1 and b for y1. |
y - y1
y - b |
= m(x - x1)
= m(x - 0) |
| Simplify the right side.
Add b to both sides. |
y - b
y |
= mx
= mx + b |
Definition —
Slope-Intercept Form for the Equation of a Line
The slope-intercept form for the equation of a line with slope m and
y-intercept (0, b) is:
y = mx + b
Example 1
Find the equation of the line with y-intercept (0, -6) and slope 2.
Write the equation in slope-intercept form.
Solution
| Substitute the given values in the
slope-intercept form of the equation.
Both the slope, m, and the y-intercept, (0, b),
are given: m is 2 and b is -6.
Simplify. |
y = mx + b
y = 2x + (-6)
y = 2x - 6 |
The slope-intercept form of the equation of the line with y-intercept (0, -6)
and slope 2 is y = 2x - 6.
Note — To Write the Equation of a Line Given Two Points
Here are two ways to write the equation of a line when given two
points:
| Use the Point-Slope Form
Step 1 Find the slope.
Step 2 Substitute the slope,
m, and the coordinates of
one of the points, (x1, y1), in
y - y1 = m(x - x1)
Step 3 Simplify. |
Use the Slope-Intercept Form
Step 1 Find the slope.
Step 2 Substitute the slope,
m, and the coordinates of
one of the points, (x, y), in y = mx + b
Then, solve the equation for b.
Step 3 Substitute the value of m
and the value b in y = mx + b. |
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