Standard Form at a Glance

Don't walk away just yet; we still need to introduce you to the second half of the deal: equations of lines in standard form. Remember? We promised a two-for-the-price-of-one special.

Sample Problem

Graph the line 2x – 3y = 4.

This equation is already in the standard form of a line. Standard form looks like this:

Ax + By = C

A, B, and C are all real numbers of some kind, but A and B are not both 0; if they were, we wouldn’t have an equation and that would be pointless.

While slope-intercept form lets us find the slope and y-intercept quickly, standard form gives us easy access to the x- and y-intercepts. Just plug in 0 for the other variable and solve. Like so:

2x – 3(0) = 4

2x = 4

x = 2

That's the x-intercept, (2, 0). Now plug in x = 0.

2(0) – 3y = 4

-3y = 4

And here's the y-intercept, . Now we have two easy-to-plot points at our disposal. No messing around with the slope, no rising and running, just two dots and a line and we're done.

Sample Problem

Find the slope of the following line, then graph it.

-2x – y = 6

Hey now, didn't we just say that we didn't have to mess with the slope anymore? What gives?

Put down the pitchforks and stop asking for a refund. Sometimes there are good reasons to find the slope of a line in standard form. For instance, what if we want to see if it's parallel or perpendicular to another line? That's a good reason, see?

There are three ways to go about finding the slope of a line in standard form. The first is to find the x- and y-intercepts, and then calculate the slope from them.

-2x – y = 6

-2x – 0 = 6

x = -3

We have (-3, 0) for the x-intercept.

-2(0) – y = 6

y = -6

And (0, -6) is the y-intercept. Now we use the formula to find the slope. This is a lot faster than checking under every rock and inside every cabinet drawer.

At least there are plenty of zeros to make the calculations simple.

Our first method found a slope of -2. The other two better find the same thing, or we'll be all kinds of grumpy.

The second method is to convert the equation to slope-intercept form. That may seem like a cop-out, but the slope-intercept form was practically custom-made for finding the slope.

-2x – y = 6

-y = 2x + 6

y = -2x – 6

We again have a slope of -2. Good, good.

The last method for finding the slope is to graph the line and find it visually. We recommend skipping this method, unless we have some other reason to graph the line. Like we do now.

We already have the intercepts, (-3, 0) and (0, -6), and the slope, -2. We're way over-prepared for graphing this line.