How To Solve Equations, Part 2
Do you know the golden rule? No, not that golden rule—I’m talking about the golden rule for solving equations. Keep on reading to find out what it is!
Jason Marshall, PhD
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How To Solve Equations, Part 2
The time has come for us to dig deeper into the process of solving algebraic equations. Haven’t we already learned how to solve equations? Well, sort of…but not really. So far we’ve just tiptoed around the problem of solving equations by learning the effective but inefficient brute force method. As you may recall, it’s not a lot of fun. Which is exactly why today is the perfect day for us to learn a better plan of attack to give you the skills you need to solve algebra problems in your future..
Expressions vs. Equations
Before we learn the secrets to solving equations, let’s take a few minutes to make sure we understand exactly what equations are. A few weeks ago, I gave you an “expressions vs. equations” brain teaser to think about in which your goal was to see how many valid and unique equations you can make from the four expressions:
So, what’s the answer? Well, each of the arrows below connect two expressions that together make a valid equation:
You’ll notice that there isn’t an arrow connecting 14 and 7. Why? Because 14 = 7 is not a valid equation! Which means that we can make a total of 5 valid equations from our 4 expressions:
- x^2 – 3x + 1 = 14
- –3x + 2 = 14
- x^2 – 3x + 1 = –3x + 2
- –3x + 2 = 7
- x^2 – 3x + 1 = 7
But wait! Aren’t there 5 more equations? Can’t we switch the expressions on the left and right side of these equations so that, for example, we turn the first equation into 14 = x^2 – 3x + 1 instead of x^2 – 3x + 1 = 14? Well, this new equation is perfectly valid, but it’s not unique. In fact, it means the exact same thing as the flipped around version.
The Golden Rule
With that out of the way, it’s time to tackle today’s big problem: solving equations. And the secret to solving equations is what I like to call the golden rule. If you learn it, live it, love it, and follow it, all will be well. If you ignore it, all will be lost. OK, perhaps things won’t be that bad—but you really do want to follow this rule! Here it is:
That which you do to one side of an equation, you must also do to the other.
It sounds simple enough, but let’s look at a few examples to make sure we fully understand what it means. To help in this endeavor, let’s once again bring out our old friend the Pythagorean Theorem. Let’s also bring out an imaginary scale to help us picture how the a^2 + b^2 and c^2 sides of the theorem balance each other out:
The basic idea conveyed by the golden rule is this: Whenever you do something to one side of an equation that changes its value (which you can think of as the weight on that side of the scale), you must do the same thing to the other side of the equation (or to the weight on the other side of the scale) to ensure that its value changes by the exact same amount. If you don’t, then the two sides of the equation will be thrown out of balance and the scale will analogously tip to one side or the other.
The Golden Rule and Arithmetic
Let’s now take a look at some examples of applying the golden rule. In particular, let’s look at how the golden rule is used with the big four arithmetic operations. After looking at the following examples, test yourself to see if you can fill in the blanks of the expressions on the right and keep the equations balanced. Check out the answers at the end of the article.
- Addition: If you add an expression, any expression—it can be a number, a variable, or some more complicated combination of numbers and variables—to one side of an equation, you must add the same expression to the other side…
- Subtraction: If you subtract an expression from one side of an equation, you must subtract the same expression from the other side…
- Multiplication: If you multiply one side of an equation by an expression, you must multiply the other side by the same expression…
- Division: If you divide one side of an equation by an expression, you must divide the other side by the same expression…
- Everything Else: If you take the square root of or raise one side of an equation to a power, you must take the square root of or raise the other side to that power, too…
How to Solve Equations
Don’t do anything that unbalances an equation.
The take home message is simple: Don’t do anything that unbalances an equation. Why? Because if you do, you’ll literally destroy the “equal” part of the equation. In other words, if you don’t treat the two sides of an equation equally, then the stuff on the left side won’t equal the stuff on the right side anymore—which means that it’s not an equation anymore!
But there’s one thing I haven’t mentioned yet—and that’s why would you ever want to do any of this stuff to equations in the first place? Are we just adding and subtracting stuff from both sides of equations for fun? While you might find it kind of fun, the truth is that there’s a much better reason for it. Namely, it allows us to solve equations. How does it work? Check back next time to find out.
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.