Do you think that solving equations is hard? Would you believe that it doesn’t have to be? Want to learn an easier way to do it? Keep on reading to find out!
We learned quite a few important things in the last episode. First, we learned that the infamous Knot Dude was once upon a time challenged to a sort of mathematical duel by a group of seafaring pyramid builders. The group’s skipper said, “If I wanted you to build me 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?”
We next learned that Knot Dude figured out that he could easily squash this challenge and humiliate his father’s pyramid building rivals if he could simply figure out how to solve the Pythagorean Theorem equation, a^2 + b^2 = c^2, for the variable b. And we learned that while doing so, Knot Dude developed an easy 4-step method—the first two steps of which we learned last time—for solving equations that’s still in use!
Today we’re going to learn the final two steps to Knot Dude’s method and we’re going to put it all together and figure out precisely how Knot Dude solved his problem and sent those nautically inclined pyramid builders sailing away in defeat..
Step 1: Simplify Each Side of the Equation
As we learned last time, the first step in solving an equation is to make the equation as simple as possible. This means that you need to start by using the golden rule of equation solving and the order of operations, PEMDAS, to make the expression on each side of the equals sign as simple as possible. In the example that we talked about last time, we added, subtracted, multiplied, and divided until we turned the equation
2 + x – 2 • 5 = 4 / 2 – x
into the very much simplified version of itself
x – 8 = 2 – x
These two equations may not look the same, but as we verified last time, they really are just different ways of writing the same underlying equation!
Step 2: Move Variable to One Side
The next step in solving an equation for a particular 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. In Knot Dude’s problem of solving for b in the Pythagorean Theorem, a^2 + b^2 = c^2, it’s most convenient for us to find a way to isolate b on the left side and move everything else to the other side.
As we learned last time, we can do that by subtracting a^2 from both sides (to keep everything balanced according to the golden rule). When we do that, we get a slightly different form of the Pythagorean Theorem equation that looks like
b^2 = c^2 – a^2
Which is great, because we’re now very close to solving for b.