Linear Equations in Point-Slope Form

Equations can be written in many forms. The previous Concepts taught you how to write equations of lines in slope-intercept form. This Concept will provide a second way to write an equation of a line: point-slope form.

The equation of the line between any two points \(\begin{align*}(x_1,y_1)\end{align*}\) and \(\begin{align*}(x_2,y_2)\end{align*}\) can be written in the following form: \(\begin{align*}y-y_1=m(x-x_1)\end{align*}\).

To write an equation in point-slope form, you need two things:

1. The slope of the line
2. A point on the line

Let's write the equation for the a line containing (9, 3) and (4, 5) in point-slope form:

Begin by finding the slope.

\(\begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}=\frac{5-3}{4-9}=-\frac{2}{5}\end{align*}\)

Instead of trying to find \(\begin{align*}b\end{align*}\) (the \(\begin{align*}y-\end{align*}\)intercept), you will use the point-slope formula.

\(\begin{align*}y-y_1& =m(x-x_1)\\ y-3& = \frac{-2}{5}(x-9)\end{align*}\)

It doesn't matter which point you use.

You could also use the other ordered pair to write the equation:

\(\begin{align*}y-5= \frac{-2}{5}(x-4)\end{align*}\)

These equations may look completely different, but by solving each one for \(\begin{align*}y\end{align*}\), you can compare the slope-intercept form to check your answer.

\(\begin{align*}y-3& = \frac{-2}{5} (x-9) \Rightarrow y=\frac{-2}{5} x+\frac{18}{5}+3\\ y& =\frac{-2}{5} x+\frac{33}{5}\\ y-5& =\frac{-2}{5} (x-4) \\ y& =\frac{-2}{5} x+\frac{8}{5}+5\\ y& =\frac{-2}{5} x+\frac{33}{5}\end{align*}\)

This process is called rewriting in slope-intercept form.