What if you were given two parallel lines in the coordinate plane? What could you say about their slopes? After completing this Concept, you'll be able to answer this question. You'll also find the equations of parallel lines and determine if two lines are parallel based on their slopes.

### Watch This

CK-12 Parallel Lines in the Coordinate Plane

### Guidance

Parallel lines are two lines that never intersect. In the coordinate plane, that would look like this:

If we take a closer look at these two lines, the slopes are both \begin{align*}\frac{2}{3}\end{align*}.

This can be generalized to any pair of parallel lines. **Parallel lines always have the same slope and different \begin{align*}y-\end{align*}intercepts.**

#### Example A

Find the equation of the line that is parallel to \begin{align*}y=-\frac{1}{3}x+4\end{align*} and passes through (9, -5).

Recall that the equation of a line is \begin{align*}y = mx + b\end{align*}, where \begin{align*}m\end{align*} is the slope and \begin{align*}b\end{align*} is the \begin{align*}y-\end{align*}intercept. We know that parallel lines have the same slope, so the line will have a slope of \begin{align*}-\frac{1}{3}\end{align*}. Now, we need to find the \begin{align*}y-\end{align*}intercept. Plug in 9 for \begin{align*}x\end{align*} and -5 for \begin{align*}y\end{align*} to solve for the *new* \begin{align*}y-\end{align*}intercept \begin{align*}(b)\end{align*}.

\begin{align*}-5 & = -\frac{1}{3}(9)+b\\ -5 & = -3 + b\\ -2 & = b\end{align*}

The equation of parallel line is \begin{align*}y = -\frac{1}{3}x-2\end{align*}.

#### Example B

Find the equation of the lines below and determine if they are parallel.

The top line has a \begin{align*}y-\end{align*}intercept of 1. From there, use “rise over run” to find the slope. From the \begin{align*}y-\end{align*}intercept, if you go up 1 and over 2, you hit the line again, \begin{align*}m = \frac{1}{2}\end{align*}. The equation is \begin{align*}y=\frac{1}{2}x+1\end{align*}.

For the second line, the \begin{align*}y-\end{align*}intercept is -3. The “rise” is 1 and the “run” is 2 making the slope \begin{align*}\frac{1}{2}\end{align*}. The equation of this line is \begin{align*}y=\frac{1}{2}x-3\end{align*}.

The lines are **parallel** because they have the same slope.

#### Example C

Find the equation of the line that is parallel to the line through the point marked with a blue dot.

First, notice that the equation of the line is \begin{align*}y=2x+6\end{align*} and the point is (2, -2). The parallel would have the same slope and pass through (2, -2).

\begin{align*}y & =2x+b\\ -2 & = 2(2) + b\\ -2 & = 4+b\\ -6 & = b\end{align*}

The equation of the parallel line is \begin{align*}y=2x+-6\end{align*}.

CK-12 Parallel Lines in the Coordinate Plane

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### Guided Practice

1. Find the equation of the line that is parallel to \begin{align*}y=\frac{1}{4}x+3\end{align*} and passes through (8, -7).

2. Are the lines \begin{align*}y=2x+1\end{align*} and \begin{align*}y=2x+5\end{align*} parallel?

3. Are the lines \begin{align*}3x+4y=7\end{align*} and \begin{align*}y=\frac{3}{4}x+1\end{align*} parallel?

**Answers:**

1. We know that parallel lines have the same slope, so the line will have a slope of \begin{align*}\frac{1}{4}\end{align*}. Now, we need to find the \begin{align*}y-\end{align*}intercept. Plug in 8 for \begin{align*}x\end{align*} and -7 for \begin{align*}y\end{align*} to solve for the *new* \begin{align*}y-\end{align*}intercept \begin{align*}(b)\end{align*}.

\begin{align*}-7 & = \frac{1}{4}(8)+b\\ -7 & = 2 + b\\ -9 & = b\end{align*}

The equation of the parallel line is \begin{align*}y = \frac{1}{4}x-9\end{align*}.

2. Both equations are already in slope-intercept form and their slopes are both \begin{align*}2\end{align*} so yes, the lines are parallel.

3. First we need to rewrite the first equation in slope-intercept form.

\begin{align*}3x+4y&=7\\4y&=-3x+7\\y&=-\frac{3}{4}x+\frac{7}{4}\end{align*}

The slope of this line is \begin{align*}-\frac{3}{4}\end{align*} while the slope of the other line is \begin{align*}\frac{3}{4}\end{align*}. Because the slopes are different the lines are not parallel.

### Explore More

Determine if each pair of lines are parallel. Then, graph each pair on the same set of axes.

- \begin{align*}y=4x-2\end{align*} and \begin{align*}y=4x+5\end{align*}
- \begin{align*}y=-x+5\end{align*} and \begin{align*}y=x+1\end{align*}
- \begin{align*}5x+2y=-4\end{align*} and \begin{align*}5x+2y=8\end{align*}
- \begin{align*}x+y=6\end{align*} and \begin{align*}4x+4y=-16\end{align*}

Determine the equation of the line that is ** parallel** to the given line, through the given point.

- \begin{align*}y=-5x+1; \ (-2, 3)\end{align*}
- \begin{align*}y=\frac{2}{3}x-2; \ (9, 1)\end{align*}
- \begin{align*}x-4y=12; \ (-16, -2)\end{align*}
- \begin{align*}3x+2y=10; \ (8, -11)\end{align*}

Find the equation of the two lines in each graph below. Then, determine if the two lines are parallel.

For the line and point below, find a parallel line, through the given point.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 3.8.