How To Convert Point Slope Form To Slope Intercept Form

There are two conventional ways of writing the equation of a straight line. One type of equation is called point-slope form, and it requires you to know (or find out) the slope of the line and the coordinates of one point on the line. The other type of equation is called slope-intercept form, and it requires you to know (or find out) the slope of the line and the coordinates of its ​y​-intercept. If you already have the point-slope form of the line, a little algebraic manipulation is all it takes to rewrite it in slope-intercept form.

Before you move on to converting from point-slope form to slope-intercept form, here's a quick recap of what point-slope form looks like:

\(y – y_1 = m(x – x_1)\)

The variable ​m​ stands in for the slope of the line, and ​x​₁ and ​y​₁ are the ​x​ and ​y​ coordinates, respectively, of the point you know. When you see a line in point-slope form with the coordinates and slope filled in, it might look something like this:

\(y + 5 = 3(x – 2)\)

Note that ​y​ + 5 on the left side of the equation is equivalent to ​y​ – ( −5), so if it helps you recognize the equation as a line in point-slope form, you could also write the same equation as:

\(y – (-5) = 3(x – 2)\)

Next, a quick recap of what slope-intercept form looks like:

\(y = mx + b\)

Once again, ​m​ represents the slope of the line. The variable ​b​ stands in for the ​y-​intercept of the line or, to put it another way, the ​x​ coordinate of the point where the line crosses the ​y​ axis. Here's an example of an actual line written out in slope intercept form:

\(y = 5x + 8\)

When you compare the two ways of writing a line, you might notice that there are some similarities. Both retain a ​y​ variable, an ​x​ variable and the slope of the line. So all you really need to get from point-slope form to slope-intercept form is a little algebraic manipulation. Consider the example given of a line in point-slope form:

\(y + 5 = 3(x – 2)\)

1. Distribute x

Use the distributive property to simplify the right side of the equation:

\(y + 5 = 3x – 6\)

2. Isolate the y Variable

Subtract 5 from both sides of the equation to isolate the ​y​ variable, which gives you the equation in point-slope form:

\(y = 3x – 11\)