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At a Glance - Factoring

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.

Sample Problem

What's the solution to the equation 2x + xyy = 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:

Sample Problem

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 

Sample Problem

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 4from 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.

Example 1

Solve the equation x + xy = 5 for x.

Exercise 1

Solve the equation x + 2 = 3y + xy – 7 for y.  

Exercise 2

Solve the equation 3x + 6y = xyz for y

Exercise 3

Solve the equation 5y + x(y + 3) = y – 2 for y

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