Lesson Objectives

- Demonstrate an understanding of slope-intercept form
- Demonstrate an understanding of slope
- Learn how to determine if two lines are parallel
- Learn how to determine if two lines are perpendicular

## How to Determine if Two Lines are Parallel, Perpendicular, or Neither

In this lesson, we will learn how to determine if two lines are parallel lines or perpendicular lines.

Example 1: Determine if each pair of lines are parallel, perpendicular, or neither. $$6x - 5y=12$$ $$12x - 10y=-15$$ Solve each for y: $$y=\frac{6}{5}x - \frac{12}{5}$$ $$y=\frac{6}{5}x + \frac{3}{2}$$ We can see that each slope of each line is 6/5. This tells us we have parallel lines. Example 2: Determine if each pair of lines are parallel, perpendicular, or neither. $$7x - 2y=5$$ $$2x + 7y=84$$ If we solve each for y: $$y=\frac{7}{2}x - \frac{5}{2}$$ $$y=-\frac{2}{7}x + 12$$ Our two slopes (7/2) and (-2/7) are not equal. Therefore, we know that we don't have parallel lines. We can multiply the slopes together to determine if we have perpendicular lines. We are looking for a product of (-1): $$\frac{7}{2}\cdot -\frac{2}{7}$$ $$\frac{\cancel{7}}{\cancel{2}}\cdot -\frac{\cancel{2}}{\cancel{7}}=-1$$ We can see that the product of the slopes is (-1). This tells us we have perpendicular lines. Example 3: Determine if each pair of lines are parallel, perpendicular, or neither. $$-8x - 3y=12$$ $$-5x + y=20$$ If we solve each for y: $$y=-\frac{8}{3}x - 4$$ $$y=5x + 20$$ Our two slopes (-8/3) and 5 are not equal. We can multiply the slopes together to determine if we have perpendicular lines. We are looking for a product of (-1): $$-\frac{8}{3}\cdot 5 ≠ -1$$ We can see the product of the slopes is not (-1), therefore, these lines are not perpendicular. We can say these two lines are not parallel lines and they are not perpendicular lines either.

Example 4: Find the equation of the line that satisfies the given conditions.

Through: $$(-3, -2)$$ Parallel to: $$x+3y=-6$$ First, let's solve the given equation for y: $$y=-\frac{1}{3}x - 2$$ From the slope-intercept form, we know the slope is -1/3. Additionally, we know that parallel lines have the same slope. We now have the slope of -1/3 and a point on the line of (-3, -2). Let's plug into the point-slope formula: $$y - y_1=m(x - x_1)$$ $$y - (-2)=-\frac{1}{3}(x - (-3))$$ $$y + 2=-\frac{1}{3}(x + 3)$$ Solve for y: $$y + 2=-\frac{1}{3}x - 1$$ $$y=-\frac{1}{3}x - 3$$ We can also place the line in standard form: $$ax + by=c$$ Here, we will use the stricter definition: $$y=-\frac{1}{3}x - 3$$ $$\frac{1}{3}x + y=- 3$$ $$x + 3y=-9$$

### Parallel Lines

Parallel lines are any two lines on a plane that will never intersect. We can determine if two lines are parallel by examining the slope of each. Two non-vertical parallel lines have slopes that are equal. We specified non-vertical here since vertical lines have an undefined slope. Let’s look at an example of parallel lines. Suppose we encounter the following two equations: $$-2x + y=5$$ $$4x - 2y=6$$ If we solve each for y: $$y=2x + 5$$ $$y=2x - 3$$ In each case, we can see that the slope is the same (2). The y-intercepts are different (0,5) and (0,-3). Since each line has the same slope or steepness, they will never touch each other. Let's look at a graph for further illustration: We can see from our graph that these two lines will never intersect.### Perpendicular Lines

Perpendicular Lines are lines that intersect at a 90° angle. Two non-vertical perpendicular lines have slopes whose product is -1. Let's look at an example of perpendicular lines. Suppose we encounter the following two equations: $$3x + 2y=4$$ $$2x - 3y=3$$ If we solve each for y: $$y=-\frac{3}{2}x + 2$$ $$y=\frac{2}{3}x - 1$$ The slope of the first equation is -3/2, while the slope of the second equation is 2/3. If we multiply the two slopes together, we get a product of (-1): $$-\frac{3}{2}\cdot \frac{2}{3}$$ $$\require{cancel}-\frac{\cancel{3}}{\cancel{2}}\cdot \frac{\cancel{2}}{\cancel{3}}=-1$$ Since our two slopes multiply together to give us a product of (-1), we know our lines are perpendicular. Let's look at a graph for further illustration: We can see from our graph that these two lines intersect at a 90° angle. Let's look at a few examples.Example 1: Determine if each pair of lines are parallel, perpendicular, or neither. $$6x - 5y=12$$ $$12x - 10y=-15$$ Solve each for y: $$y=\frac{6}{5}x - \frac{12}{5}$$ $$y=\frac{6}{5}x + \frac{3}{2}$$ We can see that each slope of each line is 6/5. This tells us we have parallel lines. Example 2: Determine if each pair of lines are parallel, perpendicular, or neither. $$7x - 2y=5$$ $$2x + 7y=84$$ If we solve each for y: $$y=\frac{7}{2}x - \frac{5}{2}$$ $$y=-\frac{2}{7}x + 12$$ Our two slopes (7/2) and (-2/7) are not equal. Therefore, we know that we don't have parallel lines. We can multiply the slopes together to determine if we have perpendicular lines. We are looking for a product of (-1): $$\frac{7}{2}\cdot -\frac{2}{7}$$ $$\frac{\cancel{7}}{\cancel{2}}\cdot -\frac{\cancel{2}}{\cancel{7}}=-1$$ We can see that the product of the slopes is (-1). This tells us we have perpendicular lines. Example 3: Determine if each pair of lines are parallel, perpendicular, or neither. $$-8x - 3y=12$$ $$-5x + y=20$$ If we solve each for y: $$y=-\frac{8}{3}x - 4$$ $$y=5x + 20$$ Our two slopes (-8/3) and 5 are not equal. We can multiply the slopes together to determine if we have perpendicular lines. We are looking for a product of (-1): $$-\frac{8}{3}\cdot 5 ≠ -1$$ We can see the product of the slopes is not (-1), therefore, these lines are not perpendicular. We can say these two lines are not parallel lines and they are not perpendicular lines either.

### Writing Equations of Parallel or Perpendicular Lines

Using what we learned in the last lesson on equations of lines, we can find the equation of a line given a parallel or perpendicular line and a point on the line. Let's look at an example.Example 4: Find the equation of the line that satisfies the given conditions.

Through: $$(-3, -2)$$ Parallel to: $$x+3y=-6$$ First, let's solve the given equation for y: $$y=-\frac{1}{3}x - 2$$ From the slope-intercept form, we know the slope is -1/3. Additionally, we know that parallel lines have the same slope. We now have the slope of -1/3 and a point on the line of (-3, -2). Let's plug into the point-slope formula: $$y - y_1=m(x - x_1)$$ $$y - (-2)=-\frac{1}{3}(x - (-3))$$ $$y + 2=-\frac{1}{3}(x + 3)$$ Solve for y: $$y + 2=-\frac{1}{3}x - 1$$ $$y=-\frac{1}{3}x - 3$$ We can also place the line in standard form: $$ax + by=c$$ Here, we will use the stricter definition: $$y=-\frac{1}{3}x - 3$$ $$\frac{1}{3}x + y=- 3$$ $$x + 3y=-9$$

#### Skills Check:

Example #1

Determine if parallel, perpendicular, or neither. $$3x - y=5$$ $$21x - 7y=-1$$

Please choose the best answer.

A

Parallel

B

Perpendicular

C

Neither

Example #2

Determine if parallel, perpendicular, or neither.

Please choose the best answer. $$7x+y=15$$ $$2x-14y=105$$

A

Parallel

B

Perpendicular

C

Neither

Example #3

Determine if parallel, perpendicular, or neither. $$13x - 5y=17$$ $$26x + 9y=22$$

Please choose the best answer.

A

Parallel

B

Perpendicular

C

Neither

Example #4

Write in standard form.

Through: $$(-3, -2)$$ Perpendicular to: $$y=-\frac{3}{2}x$$

Please choose the best answer.

A

$$2x - 3y=0$$

B

$$4x - 5y=-9$$

C

$$4x - 3y=0$$

D

$$x + y=-2$$

E

$$-2x + 3y=0$$

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