Algebra, Graphs, Lines

LINEAR EQUATIONS: They’re Like Lady Gaga

Standard Form.

Slope-Intercept Form.

Point-Slope Form.

What do they all mean, when do you use which one, and why do they each exist?

The bottom line (LOL): They’re all different names for the same thing.

Let’s think about Lady Gaga. With a new single out and a new album on the horizon, she’s already on our minds. Born “Stefani Joanna Angelina Germanotta,” she adopted the moniker “Lady Gaga” when she began her music career, and then around the release of Born This Way, began referring to herself as “Mother Monster.”

LINEAR EQUATIONS: They’re Like Lady Gaga

Later she added the androgynous persona of “Jo Calderone” to her bag of monikers. LINEAR EQUATIONS: They’re Like Lady Gaga

These are all different names, but when we hear one of them, we know who they’re referring to, and that’s the same person. Yes, ok: one could argue that “Mother Monster” is profoundly different from “Jo Calderone,” but rather than get into a heated debate about the role of gender and identity in the 21st century, our point is that these are different personas of the same human being who was born in a hospital somewhere in New York.

So let’s apply this to lines.

Here’s a line.

LINEAR EQUATIONS: They’re Like Lady Gaga

This line has 3 names:

x – 2y = -8 (Standard Form / “Stefani”)

y = 1/2x + 4 (Slope-Intercept Form / “Gaga”)

y – 2 = 1/2(x + 4) (Point-Slope Form / “Mother Monster”)

Whatever name/form you choose, if you graph it, you’re gonna get the line above. Got it?

Now.

Why are there three different names for lines? Why can’t there just be one?

Just like Gaga’s different labels, they each have different purposes. Let’s break ‘em down for you:

Standard Form: Ax + By = C

What the letters mean: 

A, B, and C are whole numbers. and y just stay there like placeholders, unless you want to plug the coordinates of a point into the equation to see if it’s on the line. Then you plug them into x and y.

Rules:

A (the first whole number, which is buddy buddy with xcan never be negative. If it is negative, you’ve gotta divide or multiply both sides of the equation by -1 to switch the signs. B and C can totes be negs.

No fractions or decimals. Only pretty and plain whole numbers like 1, 3, 7, -2 (though NOT for A, see above), etc.

When to use it:

We really only like using Standard Form to find x- and y-intercepts. To find the x-intercept, plug in 0 for x and solve for y. To find the y-intercept, plug in 0 for y and solve for x. Standard Form is set up really perfectly for this, so you don’t have to rearrange stuff.

Slope-Intercept Form: y = mx + b

What the letters mean:

stands for slope. There are a lot of theories about why it’s the letter m and not an s. If you wanna Google it, you’ll end up with a ton of different arguments and probably a headache. Our favorite way to remember it is that “mountain” starts with an m, and slope is kind of like figuring out how steep a mountain is.

stands for the y-intercept. It’s normal to think that y should stand for the y-intercept, but y hangs out most of the time in the equation, unless you want to tell if a point is on the line, and then, like we talked about above, you would plug the y-coordinate in there. Why is the y-intercept the letter b? It goes back to another equation of a line, the two-intercept form, which doesn’t get used enough to talk about, where the x-intercept is a and the y-intercept is b, prolly bc they’re the first two letters of the alphabet. The way we remember is to think of the word begin, cuz when you graph a line, you often start by plotting the y-intercept.

Rules:

Not really any, except it’s useful to make m a fraction, since the numerator reps the rise fo the slope and the denominator reps the run.

Also, always make sure y is positive and alone. Check out how to make y fly solo here.

When to use it:

Slope-Intercept Form is our most favorite for graphing. Plot the y-intercept (b), throw down your slope (m), and you’ve got a line.

Point-Slope Form: y – y1 = m(x – x1)

What the letters mean:

m still means slope. plain x and y are still placeholders / places to plug in coordinates.

x1 and y1 are the coordinates of any point on the line. any point.

Rules:

Keep plain y positive. Stay in form. Make sure you got the right slope. If you need to plug in a negative number for x1 or y1, it’ll change the sign in front of the number to a +.

When to use it:

This form is great to use whenever you get one of those problems that says: “Write the equation of a line that has a slope of 3 and passes through the point (-2, 6).” Then, all you gotta do is plug in that slope for m and the coordinates of that point for x1 and y1. Like so: y – 6 = 3(x + 2)

You can also use it for problems that say: “Write the equation of a line that passes through the points (4, 2) and (6, -3).” Then it’s a two-stepper: Use the formula for finding slope (m = y2 – y1 / x2 – x1), plug in the slope you find for m, and then pick one of the two points and plug that in. Like so: y – 2 = -2/5(x – 4)

Want more information about anything we discussed? Have a favorite form of the line and want to tell us why? Leave us a comment below!

XO, B.