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If we take the equation of a line and solve it for y we get the slope-intercept form for the equation of a line.

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 a line in slope-intercept form that has slope m = 2 and y-intercept (0, -3).

 Solution y = mx + b The point (0, -3) has the form (0, b). So, b is -3. Thus, replace m with 2 and b with -3. y = 2x - 3 The slope-intercept form for the equation of this line is y = 2x - 3

Example 2

Write the equation 4x - 3y = 12 in slope-intercept form.
 Solution 4x - 3y = 12 Solve the given equation for y. Subtract 4x from both sides. -3y = -4x + 12 Divide both sides by -3.  Simplify. y So, the equation in slope-intercept form is .

The slope of this line is and the y-intercept is (0, -4). Example 3

Find the slope-intercept form for the equation of the line that passes through the points (-6, 1) and (2, 5).

Solution

First, we need to find the slope, m. In the slope formula, let (x1, y1) = (-6, 1) and (x2, y2) = (2, 5). So, the slope is .

Note:

You can use either point (x1, y1) and (x2, y2).

 Next, use the point-slope formula. y - y1 = m(x - x1) Let (x1, y1) = (-6, 1) and y - 1 y - 1 Then, solve for y. Distribute on the right side. y - 1 Add 1 to both sides. y So, the slope-intercept form for the equation Note:

You can also write in standard form, Ax + By = C. To do this, subtract from both sides to get Now, multiply both sides by 2 to get: -x + 2y = 8 