Lesson Video: Equations of Vertical and Horizontal Lines Mathematics

Video Transcript

In this video, we’ll learn how to write the equations of vertical and horizontal lines. Remember, a horizontal line goes from left to right — think of, for example, how you view the horizon — whereas a vertical line goes straight up and down. Let’s plot a simple horizontal and vertical line and look for some patterns.

Here is a simple horizontal line. We’re going to begin by identifying the coordinates of a few points on this line. Take this point for example. Its 𝑥-coordinate is three, and its 𝑦-coordinate is two. So its coordinate pair is three, two. What about this point here? This time, the coordinate of this point is five, two. And what about a point over here? The coordinate of this point is negative three, two. We have a coordinate here that’s zero, two and a coordinate here that’s negative one, two. What do you notice about each of these coordinates?

Every single point has a 𝑦-coordinate of two. This means that we can write the equation of this line as 𝑦 equals two. In general, we say that a horizontal line that passes through the 𝑦-axis at some value 𝑎 is of the form 𝑦 equals 𝑎.

What about vertical lines? Let’s identify a few points on a single vertical line. This point has coordinates negative three, four. This point has coordinates negative three, two. The line passes through a point with coordinates negative three, negative two and another with coordinates negative three, zero. So what do we spot this time?

Well, this time, the 𝑥-coordinate remains unchanged. It’s always equal to negative three. And so we can say that the equation of this vertical line is 𝑥 equals negative three. And we can generalize and say that the equation of a vertical line that passes through the 𝑥-axis at some point 𝑏 is of the form 𝑥 equals 𝑏.

Now, a common misconception here is to think that because the line is parallel to the 𝑦-axis, it must be of the form 𝑦 equals negative three. A nice way to remember that this is not the point is to look at the axis the line passes through. This line passes through the 𝑥-axis at negative three. So its equation is 𝑥 equals negative three.

Let’s have a look at an example of this.

Which of the following represents a line parallel to the 𝑥-axis? Is it (A) negative 𝑥 minus five 𝑦 equals three, (B) negative two 𝑥 minus 𝑦 equals zero, (C) negative nine 𝑦 minus seven equals zero, or is it (D) eight 𝑥 plus three 𝑦 equals negative nine?

A line that is parallel to the 𝑥-axis is a horizontal line. It might look like this, for example. We say that the equation of a horizontal line that passes through the 𝑦-axis at some value 𝑎 is of the form 𝑦 equals 𝑎. So we’re going to rearrange each of our equations in turn to make 𝑦 the subject. Our aim is to have something of the form 𝑦 is equal to some constant.

Let’s start with the equation negative 𝑥 minus five 𝑦 equals three. We’ll begin by adding 𝑥 to both sides. So negative five 𝑦 equals three plus 𝑥. Then we divide through by negative five. And in doing so, we get the equation 𝑦 equals three plus 𝑥 over negative five or negative three minus 𝑥 over five.

Now, this quite clearly is not of the form 𝑦 equals some constant. And so we disregard equation (A). Now, in fact, when we rearrange equation (B) and equation (D), we end up with 𝑦 being some function of 𝑥 as well. This represents a diagonal line. So we can disregard (B) and (D). So that leaves us with (C). Let’s check.

We’ll begin by adding seven to both sides of this equation. So negative nine 𝑦 equals seven. Next, we divide through by negative nine to give us 𝑦 equals negative seven-ninths. This is now of the form 𝑦 is equal to some constant. And so the correct equation is (C). It’s negative nine 𝑦 minus seven equals zero.

Now, in fact, the graph with equation 𝑦 equals negative seven-ninths will pass through the 𝑦-axis at negative seven-ninths. Its 𝑦-coordinate will always be negative seven-ninths.

We’ll now look at an example that involves finding the equation of a line parallel to the 𝑦-axis.

Which of the following equations represents a line parallel to the 𝑦-axis? Is it (A) five 𝑥 minus nine equals zero, (B) negative three 𝑥 plus nine 𝑦 equals zero, (C) nine 𝑥 plus 𝑦 equals negative four, or (D) seven 𝑥 plus six 𝑦 equals four?

A line that’s parallel to the 𝑦-axis is a vertical line. It might look a little something like this. And we say that the equation of a vertical line that passes through the 𝑥-axis at some value 𝑎 is of the form 𝑥 equals 𝑎, where 𝑎 is constant. So we’re going to look at each of our equations and make 𝑥 the subject.

Let’s take equation (A). To make 𝑥 the subject, we perform a series of inverse operations. First, we want to eliminate negative nine. So we add nine to both sides. And our equation becomes five 𝑥 equals nine. Next, we divide through by five, giving us 𝑥 is equal to nine-fifths. This is indeed of the form 𝑥 equals some constant value. So we can say that 𝑎 must represent a line that’s parallel to the 𝑦-axis.

Let’s consider equations (B), (C), and (D). Each equation contains both an 𝑥 and a 𝑦. And when we rearrange them to make 𝑥 the subject, we end up with 𝑥 being some function of 𝑦. This tells us that the equations (B), (C), and (D) represent diagonal lines, not vertical ones. The correct answer is (A).

Determine the equation of the line parallel to the 𝑥-axis that passes through negative one-half, four.

A line that’s parallel to the 𝑥-axis might look a little like this. It’s a horizontal line. And we say that the equation of a horizontal line that passes through the 𝑦-axis at some constant 𝑎 is of the form 𝑦 equals 𝑎. And so one way we can establish the value of 𝑎 is to find the point at which our line passes through the 𝑦-axis.

Alternatively, we know that since it’s horizontal, all of its 𝑦-coordinates will be the same. We see that when 𝑥 is equal to negative one-half, 𝑦 is equal to four. And so all of the 𝑦-coordinates on this line must be equal to four. This means it must pass through the 𝑦-axis at four. And so its equation is 𝑦 equals four.

In our next example, we’ll look at how to write the equation of a vertical line in an alternate form, given one of its points.

Determine the Cartesian equation of the straight line passing through the point negative five, negative five and parallel to the 𝑦-axis.

A line that’s parallel to the 𝑦-axis is a vertical line like this one. We say that the equation of a vertical line that passes through the 𝑥-axis at some point 𝑎 is of the form 𝑥 equals 𝑎. Now, a common misconception here is to think that since it’s parallel to the 𝑦-axis, its equation is of the form 𝑦 equals some value. That’s not true. It passes through the 𝑥-axis at 𝑎. So its equation is 𝑥 equals 𝑎.

Now, one way we can establish the value of 𝑎 is to find the point at which our line passes through the 𝑥-axis. Alternatively, we know that since it’s vertical, all of its 𝑥-coordinates will be the same. We know that when 𝑥 is negative five, 𝑦 is negative five. So all of the 𝑥-coordinates on this line must be negative five. And that means the line must also pass through the 𝑥-axis and negative five.

We can therefore say that the equation of the line is 𝑥 equals negative five, although we can add five to both sides and alternatively write this as 𝑥 plus five equals zero. The equation of the straight line passing through the point negative five, negative five, which is parallel to the 𝑦-axis, is 𝑥 plus five equals zero.

In our final example, we’ll look at how to write the equations of a pair of horizontal and vertical lines given their point of intersection.

Find the equations of the lines parallel to the two axes that pass through negative two, negative 10.

Let’s sketch this out. Here is our point negative two, negative 10. There are a pair of lines that pass through this point which are parallel to the two axes. So we have one horizontal line and one vertical line. Remember, the equation of a horizontal line that passes through the 𝑦-axis at some value 𝑎 is of the form 𝑦 equals 𝑎. Whereas a vertical line that passes through the 𝑥-axis at some point 𝑏 has the equation 𝑥 equals 𝑏.

Our job is to find the values of 𝑎 and 𝑏. Let’s begin with our horizontal line. We know that on the horizontal line, all 𝑦-coordinates remain unchanged. Well, according to our point, when 𝑥 is negative two, 𝑦 is negative 10. So the 𝑦-coordinate will always be negative 10 on this horizontal line. It must, therefore, pass through the 𝑦-axis at negative 10. And so we can say its equation is 𝑦 equals negative 10.

Similarly, we know that for vertical lines, their 𝑥-coordinates remain unchanged. Now, according to our point, when 𝑥 is negative two, 𝑦 is negative 10. So the 𝑥-coordinates on our vertical line will always be negative two. And so the equation of our vertical line is 𝑥 equals negative two. The equations of the lines parallel to the two axes that pass through negative two, negative 10 are 𝑥 equals negative two and 𝑦 equals negative 10.

In this lesson, we learned that the 𝑦-coordinates of all points that lie on the same horizontal line must be the same. We say that if this horizontal line passes through the 𝑦-axis at some constant value 𝑎, then its equation is of the form 𝑦 equals 𝑎. Similarly, we saw that the 𝑥-coordinates of all points that lie on the same vertical line are themselves the same. And that for these vertical lines, if their 𝑥-coordinate is 𝑏, in other words, they pass through the 𝑥-axis at 𝑏, their equation is of the form 𝑥 equals 𝑏.