We know that we can graph a linear equation with two variables, such as the famous x and y, as a straight line. (By the way: that's true only because the variables don't have any exponents, but that's for another day). There are a few special equations that also produce a straight line. Let's look at this one: y = 3.
We can graph it by making a table of points.
First we pick a few random values for x, like -2, 0 , and 3. Then we plug these numbers into the equation in place of x. But wait—there is no x variable. No matter what x is, y will always be 3. Easiest substitution ever.
| x-value | y-value | point (x, y) |
| -2 | 3 | (-2, 3) |
| 0 | 3 | (0, 3) |
| 3 | 3 | (3, 3) |
Now we plot these points on a grid and get a line.
If we pick any point on the line, like the three shown, the y-coordinate will be 3. The x-coordinate will vary, but y will always be 3. That's why the equation for this line is y = 3.
Take a look at these lines:
Notice anything about all of these horizontal lines? If you answered that they're all y = something, then you're right.
Horizontal lines are all in y = A form, where A is any real number.
This is because we're graphing all points where y equals some number.
Now take a look at these lines:
These dudes are all vertical, and they're in the form x = something. Not surprisingly, this is because all points that lie on a vertical line have the same x-coordinate.
Here are some points on the line x = -1:
See how all the x-coordinates equal -1? That's always what happens with vertical lines. Vertical lines are all in x = B form, where B is any real number.
If we want to graph the line y = -2, all we need to do is plot all the points that have a y-coordinate of -2 and connect them, sort of like connect the dots.
We didn't really need to graph all of those points. Two would've been sufficient, but we like to make a point (no cheesy math pun intended).