- 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

Algebra may not be naturally pretty, but if we have the right foundation, blush, and lipstick, we can enhance its natural features. Of course, we won't be using literal makeup, since that makes the monitor all smeary.

When solving for a variable, we might wind up with a formula that involves some kind of fraction. When this result happens, make sure to give the fraction in reduced form.

Solve the equation *z* + 2*y* = 8 for *z*.

By rearranging, we find that This is a correct answer, but it's not as nice as it could be. Time to apply the mascara. Multiply the fraction by (a "clever form of 1") to find that

Ooh la la! Hey equationâ€”are you a model?

Another way to find the same answer is to divide both sides of the original equation by 2 to get

2*z* + *y* = 4

and then solve for *z*. Thankfully, we find the same answer either way. No going back to the drawing board for us.

Example 1

Solve the equation 2 |

Exercise 1

Solve the equation 3*x* + 6*y* = 27 for *x*.

Exercise 2

Solve the equation 6*x* + 2*y* = 14 for *x*.

Exercise 3

Solve the equation *y* - *x* + 3(*x* + 4) = 5*y *for *x*.