When we deal with linear equations, we sometimes come across equations that are parallel to each other or perpendicular to each other. You will see what they look like when graphed in this video lesson.
Linear equations, as the name suggests, are equations of straight lines. When you graph these types of equations, you get a straight line. Now these lines can be horizontal, vertical, or slanted. When you pair two of these equations together, you get to compare them to each other. Sometimes the equations intersect each other, other times they won't, and yet other times they may form a perfectly perpendicular intersection where each of the angles at the intersection is 90 degrees.
In this video lesson, it is the graphs of a pair of linear equations that we are considering. We will consider the case where the lines never meet (when they are parallel) and the case where the lines form a perfectly perpendicular intersection. We will see the interesting properties that each case presents. You will see how these properties actually make it easier for you to identify your equations that are parallel and your equations that are perpendicular.
We begin by recalling that our linear equations are most often written in slope-intercept form, which is y = mx + b, where m is our slope and b is our y-intercept. It is this form that will be most useful for us in determining whether two lines are parallel or perpendicular. Let's see how.
Parallel equations are equations where the lines never meet. Picture two roads that run side by side and you will have a pair of parallel lines. These roads are parallel because they never meet or cross each other. When we have a pair of parallel lines and their equations, we can easily tell that they are parallel just by comparing the equations.
You see, when two lines are parallel, their slopes will be the same and their y-intercepts will be different. So, what we are looking for is the same slope with different y-intercepts. For example, the equations y = 3x + 4 and y = 3x + 3 are parallel because their slopes are the same and their y-intercepts are different. What happens when we graph them? We get two lines that never intersect like this:
Now, what about perpendicular lines, lines that form 90 degree angles when they intersect? Picture the intersections in downtown areas of a city, and you are looking at perpendicular roads. What do we look for in this situation?
In this case, the only thing that we need to look at is the slope. Perpendicular lines have slopes that are negative reciprocals of each other. What does this mean? It means that we take one slope and divide 1 by that slope, and then we put a negative sign in front.
So, if one slope is 3, then the negative reciprocal is -1/3. The negative reciprocal of -1/3 is -(1 / (-1/3)), which becomes 3 after we evaluate it applying our basic algebra skills. So, we can say that 3 and -1/3 are negative reciprocals of each other.
Another way that we can think of negative reciprocals is that they are negative flipped versions of each other. Notice how the numerator and denominators have changed sides and we have a negative sign. So, if we see a pair of equations whose slopes are negative reciprocals of each other, then we are looking at a pair of perpendicular lines.
For example, the equations y = -4x + 2 and y = 1/4x + 2 are perpendicular to each other because of their slopes: -4 and 1/4 are negative reciprocals of each other. See how these numbers are negative and flipped versions of each other? How do these look graphed? They look like this:
Let's look at a graph and see if we can tell whether it has parallel or perpendicular lines. Are these lines parallel or perpendicular?
Are these lines parallel or perpendicular?
We begin by looking for two points on each line that we can use to find our slopes. For the red line, we see that we can use the points (0, 2) and (-3, 0). For the blue line, we can use the points (0, -2) and (2, -5). To calculate the slopes, we use what we know about finding slopes, that our slope equals the change in y over the change in x.
So, for our red line, the slope is (0 - 2) / (-3 - 0) = -2/-3 = 2/3. For our blue line, the slope is (-5 - (-2)) / (2 - 0) = (-5 + 2) / 2 = -3/2.
Hey, what do you know? These slopes are negative reciprocals of each other. They look like negative flipped versions of each other. This means that my lines are perpendicular to each other. That was pretty easy.
We have finished our lesson now. Let's review what we've learned. We've learned that linear equations are equations of straight lines. The form that is most useful for us in finding whether two linear equations are parallel or perpendicular is the slope-intercept form, which is y = mx + b. We've learned that parallel lines have slopes that are the same and the y-intercepts are different.
Perpendicular lines have slopes that are negative reciprocals of each other. We can think of a negative reciprocal as the negative flipped version. To find out whether a pair of graphed lines are parallel or perpendicular, we calculate the slopes of each line to see if they are the same or if they are negative reciprocals of each other. If they are negative reciprocals of each other, then the lines are perpendicular. If they are the same and the lines cross at different point on the y-axis, then they are parallel.
You will have the ability to do the following after watching this video lesson:
- Define linear equations and the slope-intercept form
- Differentiate between parallel and perpendicular lines using the slope-intercept formula
- Explain how to determine whether lines on a graph are parallel or perpendicular