### Learning Outcomes

- Determine whether two lines are parallel or perpendicular.
- Find the equations of parallel and perpendicular lines.
- Write the equations of lines that are parallel or perpendicular to a given line.

Parallel lines have the same slope and different *y-*intercepts. Lines that are **parallel** to each other will never intersect. For example, the figure below shows the graphs of various lines with the same slope, [latex]m=2[/latex].

All of the lines shown in the graph are parallel because they have the same slope and different *y-*intercepts.

Lines that are **perpendicular** intersect to form a [latex]{90}^{\circ }[/latex] angle. The slope of one line is the **negative** **reciprocal** of the other. We can show that two lines are perpendicular if the product of the two slopes is [latex]-1:{m}_{1}\cdot {m}_{2}=-1[/latex]. For example, the figure below shows the graph of two perpendicular lines. One line has a slope of 3; the other line has a slope of [latex]-\frac{1}{3}[/latex].

[latex]\begin{array}{l}\text{ }{m}_{1}\cdot {m}_{2}=-1\hfill \\ \text{ }3\cdot \left(-\frac{1}{3}\right)=-1\hfill \end{array}[/latex]

### Example: Graphing Two Equations, and Determining Whether the Lines are Parallel, Perpendicular, or Neither

Graph the equations of the given lines and state whether they are parallel, perpendicular, or neither: [latex]3y=-4x+3[/latex] and [latex]3x - 4y=8[/latex].

### Try It

Graph the two lines and determine whether they are parallel, perpendicular, or neither: [latex]2y-x=10[/latex] and [latex]2y=x+4[/latex].

If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line.

## Writing Equations of Parallel Lines

Suppose we are given the following equation:

[latex]y=3x+1[/latex]

We know that the slope of the line formed by the function is 3. We also know that the *y-*intercept is (0, 1). Any other line with a slope of 3 will be parallel to [latex]y=3x+1[/latex]. So all of the following lines will be parallel to the given line.

[latex]\begin{array}{lll}y=3x+6\hfill & \\ y=3x+1\hfill & \\ y=3x+\frac{2}{3}\hfill \end{array}[/latex]

Suppose then we want to write the equation of a line that is parallel to [latex]y=3x+6[/latex] and passes through the point (1, 7). We already know that the slope is 3. We just need to determine which value for *b* will give the correct line. We can begin with point-slope form of a line and then rewrite it in slope-intercept form.

[latex]\begin{array}{llll}y-{y}_{1}=m\left(x-{x}_{1}\right)\hfill & \\ y - 7=3\left(x - 1\right)\hfill & \\ y - 7=3x - 3\hfill & \\ \text{}y=3x+4\hfill \end{array}[/latex]

So [latex]y=3x+4[/latex] is parallel to [latex]y=3x+1[/latex] and passes through the point (1, 7).

### How To: Given the equation of a LINE, write the equation of a line parallel to the given line that passes through A given point

- Find the slope of the line.
- Substitute the given values into either point-slope form or slope-intercept form.
- Simplify.

### Example: Finding a Line Parallel to a Given Line

Find a line parallel to the graph of [latex]y=3x+6[/latex] that passes through the point (3, 0).

### Try It

## Writing Equations of Perpendicular Lines

We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the following line:

[latex]y=2x+4[/latex]

The slope of the line is 2 and its negative reciprocal is [latex]-\frac{1}{2}[/latex]. Any function with a slope of [latex]-\frac{1}{2}[/latex] will be perpendicular to [latex]y=2x+4[/latex]. So all of the following lines will be perpendicular to [latex]y=2x+4[/latex].

[latex]\begin{array}{lll}y=-\frac{1}{2}x+4\hfill & \\ y=-\frac{1}{2}x+2\hfill & \\ y=-\frac{1}{2}x-\frac{1}{2}\hfill \end{array}[/latex]

As before, we can narrow down our choices for a particular perpendicular line if we know that it passes through a given point. Suppose then we want to write the equation of a line that is perpendicular to [latex]y=2x+4[/latex] and passes through the point (4, 0). We already know that the slope is [latex]-\frac{1}{2}[/latex]. Now we can use the point to find the *y*-intercept by substituting the given values into slope-intercept form and solving for *b*.

[latex]\begin{array}{lllll}y=mx+b\hfill & \\ 0=-\frac{1}{2}\left(4\right)+b\hfill & \\ 0=-2+b\hfill \\ 2=b\hfill & \\ b=2\hfill \end{array}[/latex]

The equation for the function with a slope of [latex]-\frac{1}{2}[/latex] and a *y-*intercept of 2 is [latex]y=-\frac{1}{2}x+2[/latex].

So [latex]y=-\frac{1}{2}x+2[/latex] is perpendicular to [latex]y=2x+4[/latex] and passes through the point (4, 0). Be aware that perpendicular lines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature.

### Q & A

**A horizontal line has a slope of zero and a vertical line has an undefined slope. These two lines are perpendicular, but the product of their slopes is not –1. Doesn’t this fact contradict the definition of perpendicular lines?**

*No. For two perpendicular linear functions, the product of their slopes is –1. As you will learn later, a vertical line is not a function so the definition is not contradicted.*

### How To: Given the equation of a LINE, write the equation of a line Perpendicular to the given line that passes through A given point

- Find the slope of the given line.
- Determine the negative reciprocal of the slope.
- Substitute the slope and point into either point-slope form or slope-intercept form.
- Simplify.

### Example: Finding the Equation of a Perpendicular Line

Find the equation of a line perpendicular to [latex]y=3x+3[/latex] that passes through the point (3, 0).

### Try It

Given the line [latex]y=2x - 4[/latex], write an equation for the line passing through (0, 0) that is

- parallel to
*y* - perpendicular to
*y*

### How To: Given two points on a line, write the equation of A perpendicular line that passes through A Third point

- Determine the slope of the line passing through the points.
- Find the negative reciprocal of the slope.
- Substitute the slope and point into either point-slope form or slope-intercept form.
- Simplify.

### Example: Finding the Equation of a Perpendicular Line

A line passes through the points (–2, 6) and (4, 5). Find the equation of a perpendicular line that passes through the point (4, 5).

### Try It

A line passes through the points (–2, –15) and (2, –3). Find the equation of a perpendicular line that passes through the point (6, 4).

## Writing the Equations of Lines Parallel or Perpendicular to a Given Line

As we have learned, determining whether two lines are parallel or perpendicular is a matter of finding the slopes. To write the equation of a line parallel or perpendicular to another line, we follow the same principles as we do for finding the equation of any line. After finding the slope, use **point-slope form** to write the equation of the new line.

### Example: Writing the Equation of a Line Parallel to a Given Line

Write the equation of line parallel to a [latex]5x+3y=1[/latex] which passes through the point [latex]\left(3,5\right)[/latex].

### Try It

Find the equation of the line parallel to [latex]5x=7+y[/latex] which passes through the point [latex]\left(-1,-2\right)[/latex].

### Example: Finding the Equation of a Perpendicular Line

Find the equation of the line perpendicular to [latex]5x - 3y+4=0[/latex] which goes through the point [latex]\left(-4,1\right)[/latex].