Now that we know what parallel lines are, we will want to use them in applications. One such application is find lines that are parallel to existing lines.

This is done in the same way we converted from point-slope form of a line. In other words, we will need one point from the existing line, as well as the slope. Let's look at the following example

Example 1) A line passes through $LaTeX: \left(12,5\right)$ and is parallel to the graph of $LaTeX: y=\frac{2}{3}x-1$ . What equation represents the line in slope-intercept form?

Step 1 - Identify the slope of the line
The slope of the graph of $LaTeX: y=\frac{2}{3}x-1$ is $LaTeX: m=\frac{2}{3}$ . A parallel line will have the same slope.
Step 2 - Write an equation in slope of the line through $LaTeX: \left(12,5\right)$ with slope $LaTeX: m=\frac{2}{3}$ . Start with point slope form.

$LaTeX: y-y_1=m\left(x-x_1\right)$

$LaTeX: y-5=\frac{2}{3}\left(x-12\right)$ (substituting known values)
$LaTeX: y-5=\frac{2}{3}x-\frac{2}{3}\left(12\right)$ ( Distributive property)
$LaTeX: y-5=\frac{2}{3}x-8$ (Simplify)
$LaTeX: y=\frac{2}{3}x-3$ (Add 5 to both sides)

This is now in the desired form, and is our answer.

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