Factoring is another useful thing to keep in our bag of tricks—right between the levitating card and the disappearing coin—when solving equations for particular variables. While we've been solving a lot of equations by simplifying with the distributive property, sometimes we need to do things the other way around and factor instead. Sit back and take a load off, distributive property. You're on vacation.
What's the solution to the equation 2x + xy – y = 5?
This one is a little tricky. How do we get the x all by itself? We can't exactly declare a quarantine. Instead, look at the left-hand side of the equation, and factor out x:
2x + xy
x(2 + y)
Multiply this out, and you'll see that it works. We can rewrite the original equation as:
x(2 + y) – y = 5
From there, add y to both sides and divide by (2 + y) to find that:
Solve the equation xy + yz = xz for y.
Hey, where did all the numbers go? No worries: we can do this step with only variables just as easily. Again, we need to factor. If we factor y out of the first two terms, we get y(x + z) = xz, and so
Solve the equation 3x + xy = 4x – 2 for x.
We need to factor out x, but first, we get all the x terms on one side by subtracting 4x from each side:
-x + xy = -2
Now we factor out x:
x(-1 + y) = -2
Almost there. Now divide both sides by (-1 + y):
Be careful: When we factor out x from itself, we leave 1 behind. This is unfortunate, as we swore to 1 that we would never leave him behind, but it had to be done. We only hope he can forgive us.