Finding Equations of Lines

In the last lesson, we stated that linear functions are functions that can be written in the form \(f(x)=mx+b\) or \(y=mx+b\) where \(m\) and \(b\) are real numbers. Often, one wants to find the equation of a line. This is referring to the \(y=mx+b\) form.

To find the equation of a line, all one needs is two ingredients:

• A point \((x_1,y_1)\) on the line
• The slope \(m\) of the line.

The slope of the line indicates its “direction” and the point on the line essentially serves as a starting point from which you will head in the direction of the slope (think: over x then up y).

If you have a point on the line and the slope, you can use point-slope form to find the equation of any line. The point-slope form of a line is:

$$(y-y_1)=m(x-x_1)$$

• \(x_1\) is the \(x\)-value of the point on your line
• \(y_1\) is the \(y\)-value of the point on your line
• \(m\) is the slope of your line
• \(x\) and \(y\) are variables

So, all you need to do is plug all the relevant info given into point-slope form above, then solve the equation for \(y\) to get the equation of the line you are after!

Sometimes, you are given only a pair of points that your line passes through and NOT directly given the slope. We can find the slope of the line passing through these two points by calculating rise over run, or in other words, the change in y divided by the change in x from the first point to the second point.

That is, suppose you are given two points \((x_0,y_0)\) and \((x_1,y_1)\). The change in \(y\) between these points is \(\Delta y=y_1-y_0\) and the change in \(x\) is \(\Delta x=x_1-x_0\). Then the slope of the line joining these points is

$$m=\frac{\Delta y}{\Delta x}=\frac{y_1-y_0}{x_1-x_0}$$

Once you have the slope of your line, just plug that slope and the \(x\) and \(y\)-value of one your points into the point-slope form that you see above, and solve the equation for \(y\).

All the above will be demonstrated in the examples below!