4.6 Find the equation of a line  (Page 3/7)

Find an equation of a line given two points

Find an equation of a line that contains the points $\left(5,4\right)$ and $\left(3,6\right)$ . Write the equation in slope–intercept form.

Use the point $\left(3,6\right)$ and see that you get the same equation.

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Find an equation of a line that contains the points $\left(\mathrm{-3},\mathrm{-1}\right)$ and $\left(2,\mathrm{-2}\right)$ . Write the equation in slope–intercept form.

Solution

Since we have two points, we will find an equation of the line using the point–slope form. The first step will be to find the slope.

Find the slope of the line through (−3, −1) and (2, −2).
Choose either point.
Substitute the values into $y-y_1=m(x-x_1).$
Write in slope–intercept form.

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We have seen that we can use either the slope–intercept form or the point–slope form to find an equation of a line. Which form we use will depend on the information we are given. This is summarized in [link] .

Find an equation of a line parallel to a given line

Suppose we need to find an equation of a line that passes through a specific point and is parallel to a given line. We can use the fact that parallel lines have the same slope. So we will have a point and the slope—just what we need to use the point–slope equation.

First let’s look at this graphically.

The graph shows the graph of $y=2x-3$ . We want to graph a line parallel to this line and passing through the point $\left(\mathrm{-2},1\right)$ .

We know that parallel lines have the same slope. So the second line will have the same slope as $y=2x-3$ . That slope is $m_{\parallel }=2$ . We’ll use the notation $m_{\parallel }$ to represent the slope of a line parallel to a line with slope $m$ . (Notice that the subscript $\parallel$ looks like two parallel lines.)

The second line will pass through $\left(\mathrm{-2},1\right)$ and have $m=2$ . To graph the line, we start at $\left(\mathrm{-2},1\right)$ and count out the rise and run. With $m=2$ (or $m=\frac{2}{1}$ ), we count out the rise 2 and the run 1. We draw the line.