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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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