Solving Linear Equations with Literal Coefficients

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  • 0:07 Literally
  • 0:38 Literal Coefficients
  • 1:30 Solving Literal Equations
  • 3:02 Practice Problems
  • 5:10 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

In this lesson, we'll literally learn about literal coefficients. We'll look at how to solve linear equations that contain literal coefficients and practice solving several problems.


This lesson is literally the greatest thing you could be watching right now. Literally. Does it bug you when people use 'literally' when they shouldn't? I mean, it's literally the worst thing ever. Well, except for maybe a few things - I don't know, war, climate change, the lack of In-N-Out Burgers where I live. As much as it's literally misused, there are literally good uses of the word 'literal.' And, we're about to literally learn about one in algebra. Literally.

Literal Coefficients

Let's say you're walking down Algebra Street, and you bump into this: ax + 2 = b - 5. Whoa. Hold on. What are you supposed to do with all those letters? And, what kind of street is this where linear equations come to life and wander the streets? That's literally weird.

Your first thought is, 'I gotta move to a new neighborhood. Geometry Street has some cool-looking houses.' But wait, before you pack up, let's look again at this algebraic pedestrian. This equation has literal coefficients. A literal coefficient is a symbol that represents a constant, or a fixed number.

Wait - aren't variables just symbols used to represent numbers? Yes! And, literal coefficients are in many ways similar to variables. But in a linear equation, we treat literal coefficients more like numbers, and we're still trying to solve for the variable.

Solving Literal Equations

Let's look at how this literally works. Remember that stranger from Algebra Street? ax + 2 = b - 5. We just want to solve for x. And, how do we do that? We get x alone on one side of the equation.

First, subtract 2 from both sides. ax = b - 7. If that a were a number, like 7, we'd just divide by that number. We do the same thing with the literal coefficient. If we divide by a, we get x = (b - 7)/a. And, that's our answer. We can't go any further. We're basically defining x in terms of b and a.

When you think about that, since we can't do anything with those literal coefficients, there's actually less math to do. If a and b were numbers in that equation, we'd have to keep solving until we got a final number. This makes literal coefficients literally pretty cool. Note that our literal coefficients here were a and b; we usually use the letters from the beginning of the alphabet for our literal coefficients, like a, b, c, and d.

You may have seen those same letters used as ordinary variables, as in 3a = 15. So, how do we know when we have a variable and when we have a literal coefficient? Literally the easiest way is just to look at what the problem says. Problems with literal coefficients will usually say something like, 'if ax = 15, then x = what?' In this case, by the way, we'd divide both sides by a and get x = 15/a.

Practice Problems

Let's get a little practice. Here's one: 6y - c = b. We want to solve for y. First, we move that c over by adding c to both sides. Then, we just divide both sides by 6 to get y = (b + c)/6.

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