A 4-Step Guide to Solving Equations (Part 1)
Do you think that solving equations is hard? Would you believe that it doesn’t have to be? Want to learn an easier way how? Keep on reading to find out!
Once upon a time, a group of seafaring pyramid builders who were the biggest rivals to our dynamic duo of ancient Egyptian pyramid builders—Knot Dude and his father Papa Knot—proclaimed that “Any rookie pyramid builder can construct pyramids with square bases, but only a true master can create pyramids with bases shaped like rectangles!”
These antagonistic scalawags even scoffed at Papa Knot’s favorite method for creating square bases. They said that using the Pythagorean Theorem to calculate the length of the diagonal across a square pyramid’s basebefore measuring it out was “too easy” since “the equation is almost solved for you. Just plug a and b into a^2 + b^2 = c^2, and you’re done!”
See also: Why Is Algebra Useful?
At one point, they even went so far as to issue Knot Dude a direct challenge, saying: “If we were to ask you to build a small rectangular pyramid with one side that’s 15 feet long and a diagonal that’s 17 feet long, could you figure out how long the shorter side would have to be? Or is that too much work for you?”
Pretty strong words, right? Could Knot Dude do it? Of course he could…he was the Knot Dude, after all. So how did he do it? What did it have to do with the golden rule for equation solving that we learned last time? And how can you use Knot Dude’s easy 4-step method for solving equations? Stay tuned because those are exactly the questions we’ll be answering today.
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Recap: The Golden Rule
That which you do to one side of an equation, you must also do to the other.
Remember that rule? I hope so because, as we learned about last time, this golden rule of equation solving is going to be super important in all of your future equation solving endeavors. And it was super important for Knot Dude, too. He knew that he could solve his nemesis’ puzzle by somehow using a and c in the Pythagorean equation to solve for b. (If you don’t see why, I encourage you to stop for a minute and think about it!)
And not only that, the golden rule was also the key idea that helped Knot Dude discover his 4-step method for solving equations…exactly the method we’ll be learning over the next few weeks. Speaking of which, the first step of that method is…
Step 1: Simplify Each Side of the Equation
The first step in solving an equation is to make the equation as simple as possible. We’ll return to the Pythagorean equation—and Knot Dude’s challenge from his rival pyramid builders—in a minute. But first, to see what I mean by “simple,” let’s take a look at a different equation that needs a lot of simplifying. Namely,
2 + x – 2 • 5 = 4 / 2 – x
There are several parts of the expressions on both the left and right sides of this equation that we can simplify according to the rules laid out by the order of operations—aka, PEMDAS. First, we can perform the multiplication on the left side to find that
2 + x – 2 • 5 = 2 + x – 10
Then, if we swap around the first two pieces of this new expression on the left side, we’ll find a subtraction problem that we can do:
2 + x – 10 = x + 2 – 10 = x – 8
That left side is now a whole lot simpler, right? On the right side of the equation, 4 / 2 – x, we can perform the division problem to get
4 / 2 – x = 2 – x
When we combine these two simplified expressions for the left and right sides of the equation, we get a simplified form of the original equation that looks like
x – 8 = 2 – x
This new equation is equivalent to the first one (meaning we’ve been careful not to throw anything out of balance per the golden rule), but it’s much, much simpler. In case you’re wondering why we didn’t use the Pythagorean Theorem as our example, well, as you can verify yourself, it’s already as “simple” as it can get!
Step 2: Move Variable to One Side
After simplifying as much as you can, the next step in solving an equation for a variable is to use addition and/or subtraction to move every part of the equation that contains the variable you’re solving for to one side of the equals sign or the other. In Knot Dude’s problem, he needs to solve for b in the Pythagorean Theorem, a^2 + b^2 = c^2. Since b is on the left side of the equation, let’s focus on keeping it there and getting rid of everything else from that side of the equation.
How can we do that? Well, we now know that we’re free to subtract an expression from one side of the equation as long as we subtract the same expression from the other side of the equation, too (that’s the whole point of the golden rule). So, let’s get rid of the a^2 on the left side of the Pythagorean equation by subtracting it from both sides:
a^2 + b^2 – a^2 = c^2 – a^2
Now, let’s swap the first two pieces of the expression on the left to find that
a^2 + b^2 – a^2 = b^2 + a^2 – a^2
which we can simplify to
b^2 + a^2 – a^2 = b^2
The end result is that we’ve turned the Pythagorean Theorem equation into
b^2 = c^2 – a^2
Remember, Knot Dude’s goal for this step was to isolate the variable b on one side of the equals sign, and that’s exactly what he—and we—have managed to do!
Steps 3 and 4
Which means that it’s time for the third step—isolate the variable (via multiplication, division, exponentiation, or taking roots)—as well as the surprise fourth and final step.
But, unfortunately, that’s all the math we have time for today. So the big reveals about how to carry out the aforementioned third step, and exactly what the mysterious fourth step is, will have to wait until next time. As will the exciting conclusion to the tale of Knot Dude, Papa Knot, and their no-good seafaring pyramid building rivals. So be sure to check back and find out how it all turns out!
If you want to learn more about algebra, please check out my book The Math Dude’s Quick and Dirty Guide to Algebra.
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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!