Equations and Inequalities
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.
Solve 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 = (2 + y)x
Multiply this out, and you will see that it works. We can rewrite the original equation as
(2 + y)x - 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 to find that
x( - 1 + y) = - 2
and divide by ( - 1 + y) to find that
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.