Parallel and Perpendicular Lines Lesson

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

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: