How To Solve Equations (Part 1)
How should you think about equations? And how can you solve them? Keep on reading to find out!
In the last episode of Math Dude, we learned that an equation is a proclamation that the expression on one side of an equals sign has the exact same value as the expression on the other. But what exactly does that mean? And how do you use equations to solve problems and do useful things? Stay tuned because those are precisely the questions we’ll be answering today and over the next few weeks.
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How to Solve an Equation
When you plug in just the right number for the variable, the expressions on either side of the equals sign in the equation
must have the exact same value. After all, that’s what an equation means! But how do you find that special value—the “solution” to the equation? As we’ll see over the next few weeks, there are actually several different ways to do it. The method we’re going to talk about today is by no means the best for most situations, but it’s nonetheless a helpful one to keep in the back of your mind. This method is the most rudimentary of all methods—we’re talking about good old brute force.
What does that mean in terms of math? It means that we start by guessing that the value of the variable must be equal to 1, and we then calculate the values that the left and right expressions will have. If the two expressions are equal, we’re done. If not, we move on to the next guess and calculate the values of the two sides when the variable is equal to 2. We then check again and we repeat as necessary until we find the value of the variable that gives us a match.
Here are the values of the expressions on the left and right side of our equation for guesses ranging from 1 to 5:
For guesses of 1, 2, and 3, the value of the left expression is greater than the value of the right. For a guess of 5, the value of the left expression is less than the value of the right. And, lo and behold, the left and right expressions have the exact same value for a guess of 4. Which means that this equation is solved by setting the variable to 4.
The Problem with Brute Force
While it’s wonderful that we solved this problem, it’s clear that this kind of approach isn’t going to cut it in the long run. While it works just fine for simple cases, there are a lot of situations where a brute force approach to solving algebra problems can leave you with a headache. For example, what if the solution was a huge number—it would’ve taken forever to go through all the possible guesses to find the right one!
Which means that we need to find a better way. And, as we’ll soon see, that’s precisely where the methods of algebra that people have developed over the past centuries come to the rescue.
How To Think About Equations
We’ll get to those more efficient methods for solving algebra problems soon, but first I want to talk about exactly what an equation means. In particular, I want to paint a picture in your mind that will help you think about the meaning of equations. The trick is to think of an equation as an old-fashioned balance scale. This scale takes the stuff on the left and right sides of an equals sign and checks to see if they balance.
When the two sides of the scale do balance for some particular values of the variables, the equality is true and the equation is therefore valid for those values of the variables. But if one side is larger than the other, the scale falls out of balance and the equation is invalid. This is actually a rather powerful metaphor because it not only gives us a way to think about the meaning of an equation, as we’ll see next time it also shows us what we can and cannot do to an equation to solve it.
Expressions vs. Equations
Before finishing, I want to leave you with a brain teaser to kick around. The goal is to see how many valid and unique equations you can create from the four expressions:
By the way, you may have noticed that we’ve abandoned our beloved “box” variable. It’s true, the time has come to move on to using more grown up variables…like x. Why? There’s really not a great reason. But the truth is that you’ll be seeing a lot more of this little fella, x, in the future—so you may as well get used to it!
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 . 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.
