What are equations? How are they related to the algebraic building blocks we’ve been learning about? And what exactly do equations mean? Math Dude has the answers!
Now that we’ve learned about the meaning of algebra, the variables we use in algebra, and the expressions we write with those variables, we’re ready to move on to the main-event we’ve been building towards: equations! What are they? How are they constructed from our algebraic building blocks? And what exactly do they mean? Stay tuned because those are exactly the questions we’ll be answering today.
From Expressions to Equations
In our first episode on variables, we chose to use a symbol that looks like a box as our variable, and we discovered that we can do anything with our variable that we can do with numbers. We started by adding our box to the number 3:
We then imagined that our variable is no longer an empty box but is instead a box that contains the number 2. When we wrote this down, we arrived at something that looks very familiar—an everyday arithmetic problem:
The only thing that makes this look a little unusual is the outline of a box reminding us that the number 2 is actually the value of a variable.
But now let’s do something that’s really unusual—let’s pretend that we don’t remember setting our variable equal to 2 to come up with this arithmetic problem. In fact, let’s pretend that we don’t actually remember coming up with the problem ourselves at all. Instead, let’s imagine that we stumbled upon the very similar but oh-so-profoundly different problem:
What’s the big deal here? Well, if you think about it, you’ll see that this is an entirely new kind of problem—it’s a problem that’s asking you to figure out what value a variable must have in order to make some statement true. In other words, what value must our variable be equal to so that adding it to 3 gives you 5?
In algebra, this type of statement goes by another name—it’s called an equation. While it isn’t particularly tough to figure out what the answer must be in this case, this problem nonetheless contains the true essence of algebra. By which I mean that it takes us from a fragmented world containing nothing but mathematical phrases, to a rich and brave new world full of mathematical sentences. In other words, it takes us from expressions to equations.