# 4.6.2: Equations of Parallel Lines

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- 4323

## Equations of Parallel Lines

Suppose a coordinate plane is transposed over the map of a city, and Main Street has the equation y=2x+5. If Broad Street runs parallel to Main Street, and it passes through the point (3, 14), what would be the equation of Broad Street? How do you know?

**Equations of Parallel Lines**

Recall that **parallel lines** have the same slope.

Each of the graphs below have the same slope, which is 2. According to the definition, all these lines are parallel.

**Let's determine if y=1/3x−4 and −3x+9y=18 are parallel:**

The slope of the first line is 1/3.

Any line parallel to this must also have a slope of 13.

Find the slope of the second equation: A=−3 and B=9.

slope=−A/B=3/9→1/3

These two lines have the same slope so they are parallel.

**Writing Equations of Parallel Lines**

Sometimes, you will asked to write the equation of a line parallel to a given line that goes through a given point. In the following example, you will see how to do this.

**Let's find the equation of the following lines:**

- The line parallel to the line y=6x−9 passing through (–1, 4).

Parallel lines have the same slope, so the slope will be 6. You have a point and the slope, so you can use point-slope form.

y−y_{1}=m(x−x_{1})

y−4=6(x+1)

You could rewrite it in slope-intercept form:

y=6x+6+4

y=6x+10

- The line parallel to the line y−5=2(x+3) passing through (1, 1).

First, we notice that this equation is in point-slope form, so let's use point-slope form to write this equation.

y−y1=m(x−x1) Starting with point-slope form.

y−1=2(x−1) Substituting in the slope and point.

y−1=2x−2 Distributing on the right.

y−1+1=2x−2+1 Rearranging into slope-intercept form.

y=2x−1

**Examples**

Example 4.6.2.1

Earlier, you were told that a coordinate plane is transposed over the map of a city, and Main Street has the equation y=2x+5. If Broad Street ran parallel to Main Street and it passes through the point (3, 14), what would be the equation of Broad Street?

**Solution**

Since Broad Street is parallel to Main Street, the line representing the two streets have the same slope. The slope of the equation for Main Street is 2 so the slope of the equation for Broad Street is also 2. Now, we can use the slope and the point (3, 14) to find the equation for Broad Street.

y−y_{1}=m(x−x_{1}) Starting with point-slope form.

y−14=2(x−3) Substituting in the slope and point.

y−14=2x−6 Distributing on the right.

y−14+14=2x−6+14 Rearranging into slope-intercept form.

y=2x+8

Example 4.6.2.2

Find the equation of the line parallel to the line 4x−y=24 passing through (3, 2).

**Solution**

Since this is in standard form, we must first find the slope. For Ax+By=C, recall that the slope is m=−AB. Since A=4 and B=−1:

m=−A/B=−4/−1=4

Now that we have the slope, we can plug it in:

y−y_{1}=m(x−x_{1}) Starting with point-slope form.

y−2=4(x−3) Substituting in the slope and point.

y−2=4x−12 Distributing on the right.

y−2+2=4x−12+2 Rearranging into slope-intercept form.

y=4x−10

**Review**

- Define parallel lines.

In 2-6, determine the slope of a line parallel to each line given.

- y=−5x+7
- 2x+8y=9
- x=8
- y=−4.75
- y−2=(1/5)(x+3)

In 7-10, find the line parallel to it through the given point.

- y=−(3/5)x+2;(0,−2)
- 5x−2y=7;(2,−10)
- x=y;(2,3)
- x=−5;(−2,−3)

**Mixed Review**

- Graph the equation 2x−y=10.
- On a model boat, the stack is 8 inches high. The actual stack is 6 feet tall. How tall is the mast on the model if the actual mast is 40 feet tall?
- The amount of money charged for a classified advertisement is directly proportional to the length of the advertisement. If a 50-word advertisement costs $11.50, what is the cost of a 70-word advertisement?
- Simplify √(112).
- Simplify √(12
^{2}-7^{2}). - Is √(3)−√(2) rational, irrational, or neither? Explain your answer.
- Solve for s: 15s=6(s+32).

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.7.

### Additional Resources

PLIX: Play, Learn, Interact, eXplore: **Equations of Parallel Lines: Exploring Equations**

Video:

Activity: **Equations of Parallel Lines Discussion Questions**

Study Aid: **Determining the Equation of a Line Study Guide**

Practice: **Equations of Parallel Lines**

Real World Application: **Parallel & Perpendicular Lines**