Analytical geometry is usually introduced in high school. In analytical geometry, two lines are drawn on a paper at “right angles” to each other. The vertical line represents the “y-coordinate axis,” and the horizontal line represents the “x-coordinate axis.”¹ Using these two axes, every point on the paper can be given a value that determines where it is. If the place where the two lines cross is considered the zero point or origin, its coordinates or value (x, y) would be simply, (0, 0).
Along the horizontal x-axis, starting to the right of the (0, 0) point, write little numbers like a ruler has, 1, 2, 3, and so forth. To the left of that point, write, -1, -2, -3, and so on. For the y-axis, the 1, 2, 3, and such go upward, whereas the -1, -2, -3 and the rest go downwards.
Equation of a Line – Examples
Now we’ve prepared our coordinate system, which is what we call this. Using this coordinate system, we will draw a mathematical representation of a simple line equation. What does an equation for a line look like? Before we present the general equation of a line, we will provide three simple examples of line equations. First, the line,
x = 1
This equation means that no matter what value y has, x has the value zero. Let’s draw some points to demonstrate what we mean. We’ll pick y = 0, y = 2, y = 5, y = -3. Then, we get
Locating these on our coordinate axes, we see that all we are doing is drawing a line parallel to the y-axis, but to the right one notch!
Correspondingly, if we next consider,
y = 1
we end up with a line one notch above the x-axis!
For our final example of a line. Let’s choose, y = 2x. Some example points are,
(-2.5, 5) (Note: Yes, we can plot fractions.)
This line goes from the bottom left-hand side of the coordinate axes up to the right-hand side. The line is just a bit more vertical than horizontal. This is because we included the number 2 in the equation. If we had written y = 1/2 x instead, the line would have been a little more horizontal than vertical. It is for good reason that the number in front of the x is called the “slope.” As in skiing, the number determines the slope of the line.
General Equation for a Line
Ok. We’re ready to consider the general equation for a line. It is,
y = mx + b
We’ve already seen that m is a number representing the slope of the line. What, then, is b? It determines where the line crosses the y-axis. To demonstrate that, choose x = 0. Then if b = 2.7, for instance, y = 2.7. The line crosses the y-axis at (0, 2.7). This is the reason why b is called the “intercept.” It represents the point where the line intercepts the y-axis.
1 To see graphic representation of what is written here, see the image associated with this article.Only the upper right quarter or “quadrant” in which x and y are both positive is drawn.