What Are Equations?
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.
What Are Equations?
And with that we’ve arrived at probably the single most important idea in all of algebra—equations. In many ways, this idea isn’t new to you. In fact, you’ve been using equations since you learned basic arithmetic. After all, a simple problem like 3 + 2 is really just a mathematical expression. And, unbeknownst to you, way back in the day when you were emphatically proclaiming that the answer to 3 + 2 is 5, you were also effectively writing an equation saying that 3 + 2 = 5.
But what do equations like this really mean? Well, let’s think about perhaps the simplest equation in the world:
1 = 1
Yes, that really is a legitimate equation—it has two expressions located on either side of an equals sign (which is all that an equation needs). What does it mean? The quick and dirty answer is that equations exist solely to boldly proclaim that the expression on one side of an equals sign has the exact same value as the expression on the other. In the case of the equation 1 = 1, this means that the number 1 has to be equal to the number 1. Which, of course, it most certainly is.
What Do Equations Mean?
So that’s the simplest equation in the world, now let’s now take one step up the complexity ladder and look at an equation that’s a bit less straightforward:
As you can see, this equation also contains two expressions located on the left and right sides of an equals sign…as all equations must. But this time both expressions include numbers, variables, and operators. The two expressions look quite different, but the fact that they’re on opposite sides of an equals side means that they must evaluate to the exact same value when you plug in the correct number for the variable. Of course, that means that you can’t just plug in any old number for the variable…there must be a certain special value that does the trick. And the goal of algebra is to find that value.
How do we do it? Well, that special value is called the solution to the equation, and the process of finding the solution is the process of solving the equation. Which is exactly the topic we’ll be talking about next time.
Wrap Up
Okay, that’s all the math we have time for today. If you want to learn more about algebra, please check out my book The Math Dude’s Quick and Dirty Guide to Algebra.
Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via Facebook, Twitter, or email at mathdude@quickanddirtytips.comcreate new email.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
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About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.
