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!
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.