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 Lines or Perpendicular Lines

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

-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.

3x + 2y = 4

2x - 3y = 3

If we solve each for y:

y = -3/2 x + 2

y = 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 = 6/5 x - 12/5

y = 6/5 x + 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 = 7/2 x - 5/2

y = -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 = -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.

### 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 = -3/2 x + 2

y = 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 = 6/5 x - 12/5

y = 6/5 x + 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 = 7/2 x - 5/2

y = -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 = -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.

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