High School Algebra I Unlocked (2016)
Chapter 5. Systems of Equations and Inequalities
Lesson 5.2. Parallel and Perpendicular Lines
At the beginning of the chapter, we said that there are three possible solutions to a system of equations: multiple solutions, exactly one solution, or no solutions. Up until now, we’ve focused on systems of equations that have exactly one solution and those that have an infinite number of solutions. We haven’t, however, discussed systems of equations in which the lines never intersect, or systems of equations in which the lines intersect to form a 90-degree angle. In systems where the lines never intersect, the lines are parallel to one another; lines that intersect to form a 90° angle are perpendicular to one another.
Parallel lines have
equal slopes and never
intersect. Perpendicular
lines have slopes that are
the negative reciprocals
of one another and
intersect at a 90° angle.
Parallel and perpendicular lines are of particular interest to math folks because of the relationships between the slopes of the lines. Lines that are parallel to one another have identical slopes, never intersect, and have no solution. For example, if the slope of line is 3, all lines parallel to the original line will have a slope of 3. Alternatively, lines that are perpendicular to one another have slopes that are the negative reciprocals of one another. For example, if the slope of a line is 3, a line perpendicular to the original line will have a slope of −1/3.
Take a look at the following graphs of parallel and perpendicular lines:
Let’s try a few questions that deal with parallel and perpendicular lines.
EXAMPLE
What is the slope of a line parallel to 4x + 12 = 2y ?
Recall from our lesson on two-variable linear equations that the slope of a line is the value m when equations are written in slope-intercept form y = mx + b.
Before you can identify the slope, you must put the equation into slope-intercept form. Here, you need to divide both sides of the equation by 2.
2y = 4x + 12
y = 2x + 6
Therefore, the slope of the given line is 2. However, the question asks you to find the slope of a line parallel to 4x + 12 = 2y. Since parallel lines have equal slopes, the slope of a line parallel to 4x + 12 = 2y is also 2.
That wasn’t too bad, right? Let’s try another one.
Since this question wants you to find the value of b, start by isolating a in each equation.
a + 3b = 27 |
a − 3b = 9 |
a = 27 − 3b |
a = 9 + 3b |
Now that a is isolated both equations, you can set them equal to one another to solve for b.
27 − 3b = 9 + 3b
27 − 9 = 3b + 3b
18 = 6b
3 = b
Therefore, the correct answer is (F).
EXAMPLE
What is the slope of a line perpendicular to 3y = 3x + 9 ?
Here, you’re asked to find the slope of a line perpendicular to 3y = 3x + 9. Start by dividing both sides of the equation by 3.
3y = 3x + 9
y = x + 3
Now that the equation is in slope-intercept form, y = mx + b, we can determine that the slope of the line, m, is 1. Here, however, the question asks you to find the slope of a line perpendicular to 3y = 3x + 9. Since perpendicular lines have slopes that are the negative reciprocals of each other, the slope of a line perpendicular to 3y = 3x + 9 is −1.
Now, let’s look at a question that deals with parallel and perpendicular lines, but in a slightly different way.
EXAMPLE
If line A passes through points (3, 8) and (−1, 0), and line B passes through (8, −4) and the origin, are these lines parallel, perpendicular, or neither?
Here, you are asked to determine if line A and B are parallel, perpendicular, or neither. Find the slope of line A by substituting the points (3, 8) and (−1, 0) into the following formula:
Here is how you may see systems of equations on the Grid-In math section of the SAT.
If 4x + 2y = 24 and = 7, what is the value of x ?
Now find the slope of line B by substituting points (8, −4) and (0, 0) into the slope formula:
So the slope of line A is 2, the slope of line B is −1/2, and, because the slopes are negative reciprocals of each other, the lines are perpendicular to one another.
You should also know how to identify graphs that show parallel or perpendicular lines. Let’s see how this concept might be tested.
EXAMPLE
Which of the following lines is the graph of a line perpendicular to the line defined by the equation 3x − 7y = 28 ?
A)
B)
C)
D)
To solve this problem, begin by rewriting the equation in slope-intercept form:
3x − 7y = 28
7y = 3x − 28
y = x − 4
Therefore, the slope of the original line is . In order for a line to be perpendicular, it must have a slope that is the negative reciprocal of the slope of the given line. Thus, the slope of the line perpendicular to 3x − 7y = 28 is −. Eliminate (A) and (B), as the graphs shown have positive slopes. Now, you can use to find that the graph shown in (C) has a slope of −, and the graph shown in (D) has a slope of −. Therefore, the correct answer is (D).
Even though this is not a multiple-choice question, approach it the same way we’ve approached the other questions in this chapter. Since this question specifically asks you to find the value of x, start by rearranging the equations to isolate the y-variable.
4x + 2y = 24 |
= 7 |
2y = −4x + 24 |
7y = 7(2x) |
y = −2x + 12 |
7y = 14x |
y = 2x |
Now that you’ve isolated y in both equations, set them equal to one another to solve for x:
−2x + 12 = 2x
4x = 12
x = 3
Thus, the value of x = 3.