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.
Solve the equation x + xy = 5 for x.
Solve the equation x + 2 = 3y + xy - 7 for y.
Solve the equation 3x + 6y = xyz for y.
Solve the equation 5y + x(y + 3) = y - 2 for y.