• Find an equation of the line given the slope and $y$-intercept
  • Find an equation of the line given the slope and a point
  • Find an equation of the line given two points
  • Find an equation of a line parallel to a given line
  • Find an equation of a line perpendicular to a given line

Assignment

Find the Equation of a Line

So, you have some information about a line, but need an equation. There are various situations, but we’ll boil it down to just two.

When You Have Slope and Intercept

You have the rate of change of a function, and you know the value when $x=0$. This means you have a slope and a $y$-intercept. Use slope-intercept form.

For Every Other Situation

Do you remember the equation for slope?

\[\begin{align} m = \frac{y_1 - y_2}{x_1 - x_2} \end{align}\]

I’m going to change it slightly.

\[\begin{align} m = \frac{y - y_1}{x - x_1} \end{align}\]

And now solve it for $y$.

\[\begin{align} m &= \frac{y - y_1}{x - x_1} \\ m(x - x_1) &= y - y_1 \\ m(x - x_1) + y_1 &= y \\[1em] y &= m(x - x_1) + y_1 \label{point-slope} \end{align}\]

That equation at $\ref{point-slope}$ is point-slope form, and is a perfectly acceptable version of a linear equation. Like slope-intercept form, it requires only two things to create: a point and a slope. So, give the point $(2,5)$ and a slope of $\frac{1}{4}$, we get an equation of ${y=\frac{1}{4}(x-2)+5}$. If you are asked to write it in slope-intercept form, then just distribute and add.

\[\begin{align} y &= \frac{1}{4}(x - 2) + 5 \\ y &= \frac{1}{4}x - \frac{1}{2} + 5 \\ y &= \frac{1}{4}x + \frac{9}{2} \end{align}\]

But what if you only have two points? Well, if you have two points you can find the slope. Then you have slope and your choice of points to use.