Mathematics » Graphs and Equations » Find the Equation of a Line

# Finding an Equation of a Line Parallel to a Given Line

## Finding 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 $$\left(-2,1\right)$$.

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 $$\left(-2,1\right)$$ and have $$m=2$$. To graph the line, we start at$$\left(-2,1\right)$$ 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.

Do the lines appear parallel? Does the second line pass through $$\left(-2,1\right)$$?

Now, let’s see how to do this algebraically.

We can use either the slope–intercept form or the point–slope form to find an equation of a line. Here we know one point and can find the slope. So we will use the point–slope form.

### Example: How to Find an Equation of a Line Parallel to a Given Line

Find an equation of a line parallel to $$y=2x-3$$ that contains the point $$\left(-2,1\right)$$. Write the equation in slope–intercept form.

### Solution

Does this equation make sense? What is the y-intercept of the line? What is the slope?

### Find an equation of a line parallel to a given line.

1. Find the slope of the given line.
2. Find the slope of the parallel line.
3. Identify the point.
4. Substitute the values into the point–slope form, $$y-{y}_{1}=m\left(x-{x}_{1}\right)$$.
5. Write the equation in slope–intercept form.

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