Print This Page
**Solving Equations For Expressions**: At a Glance

- Topics At a Glance
- Solutions to Equations
- Checking Solutions to Equations
- Number of Solutions to an Equation
- Equivalent Equations
- Solving Equations with One Variable
- Adding and Subtracting Constants
- Checking Answers
- Adding and Subtracting Variables
- Multiplication and Division
- Complicated Equations
- Simplifying Equations
- Eliminating Fractions
- Keeping Both Solutions
- When You Get Stuck
**Solving Equations with Multiple Variables****Solving Equations for Expressions**- Keeping Answers Pretty
- Factoring
- Geometry
- Single-Variable Inequalities
- Strict Inequalities
- Equivalent Inequalities
- Inequalities that Allow Equality
- Solving Inequalities
- In the Real World
- Fitting Things in Spaces
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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

Solve the equation *A* = π *r*^{2} for *r*^{2}.

Divide both sides by π to find that .

Because we have *r*^{2} all by itself on one side of the equation, and no copies of *r* on the other side, that is it. Gosh, it seems like so little work for such an ugly-looking expression. Good thing we're all about inner beauty here.

Solve the equation *y* = 2*x* - 9 for 2*x*.

Add 9 to both sides to find that *y* + 9 = 2*x*. We're already done. Whoa, we barely started! If we were solving for *x*, we would divide both sides by 2, but since we are solving for 2*x,* 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 are solving for an expression involving *x*—say, *x*^{2}—we want to end up with a formula for *x*^{2} that doesn't have any *x* terms in it. Having a formula such as

isn't helpful, because if we knew *x* then we would know *x*^{2} without needing to use some fancy formula. Even though we would probably try to figure out the fancy formula before realizing that. Doh.

**Be Careful.** Remember to answer the question that is asked. If you are told to solve for 2*x*, 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

Solve for 3 9 |

Exercise 1

Solve the following equation for *z*^{3}:

*z*^{3} + 3(*z* + 2) - 4 = 2*z* + 4(*z* + 1) - 3*z*

Exercise 2

Solve for *x*^{5}:

*x*^{2}(*x*^{3} + 2) - 2*y* = *y* + 2*x*^{2}

Exercise 3

Solve for 2*y*:

4*y* + 16 = 8*x* - 3