At a Glance - Solving Equations for Expressions

We can also solve equations for expressions that aren't single variables. We can solve an equation for x2, or 2x, for example. We can sense your eager anticipation. We won't make you wait any longer.

Sample Problem

Solve the equation A = πr2 for r2.

Divide both sides by π to find that .

Because we have r2 all by itself on one side of the equation, and no copies of r on the other side, that's it. Gosh, it seems like so little work for such an ugly-looking expression. Good thing we're all about inner beauty here.

Sample Problem

Solve the equation y = 2x – 9 for 2x.

Add 9 to both sides to find that y + 9 = 2x. We're already done. Whoa, we barely started! If we were solving for x, we would divide both sides by 2, but since we're solving for 2x, we don't even need to bother. We can use all that extra time we saved to volunteer at a homeless shelter. Or...to play another level of Call of Duty.

When we're solving for an expression involving x—say, x2—we want to end up with a formula for x2 that doesn't have any x terms in it. Having a formula like isn't helpful, because if we knew x then we would know x2 without needing to use some fancy formula. Even though we'd probably try to figure out the fancy formula before realizing that. Doh.

Be Careful. Remember to answer the question that's actually being asked. If you're told to solve for 2x, don't do extra work to solve for x if you don't need to. Your time is precious. Time is money. That's money in the bank. You can take that to the bank. We think you see where we're going with this train.

Example 1

 What do we get when we solve for 3x in the equation 9x + 4y = y2 – 2?

Exercise 1

Solve the following equation for z3:

z3 + 3(z + 2) – 4 = 2z + 4(z + 1) – 3z

Exercise 2

Solve for x5

x2(x3 + 2) – 2y = y + 2x2

Solve for 2y:

4y + 16 = 8x – 3