- 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

Aw...what did they ever do to you?

When lots of fractions are involved, there is another way to make an equation look simpler before solving it: get rid of the fractions. Sweep them away, pack them in garbage bags and dump then into the bay. But not really, because that's littering. To get rid of the fractions, we pick a useful number and multiply both sides of the equation by that number. The number is useful if multiplying eliminates all fractions.

Plus, if the number does a good enough job cleaning up the fractions, maybe we'll see how it does with our bedroom.

Solve the equation .

**Way 1.** Subtract 2/3 from each side, so that , and add the fractions.

**Way 2.** Find the LCD of the fractionsâ€”in this case, 6. Multiply the left-hand side of the equation by 6 and the right-hand side of the equation by 6 to find that

Oy. So many fractions and parentheses, we better simplify this sucker. Fortunately, it simplifies to

4 + 6*x* = 1

Much better. Notice that there are no longer fractions in the equation, not to mention that the parentheses are gone as well, which is a nice bonus. Subtract 4 from both sides, and we see that

6*x* = - 3

so is the solution to the equation. We wound up with a fraction anyway, but it sure was nice being without them at least for a little while.

Make sure you understand that getting rid of fractions isn't the same thing as "simplification." When we "simplify," we rewrite the expressions on each side of the = sign to be tidier, but we don't change the value of either expression. When we eliminate fractions, we are multiplying both sides of the expression by the same number and therefore changing the values of both expressionsâ€”but in such a way that the scale is still balanced. Each side is much, much heavier. In fact, we should probably put the whole thing on a sturdier table.

Example 1

Solve the equation |

Exercise 1

Solve the equation .

Exercise 2

Solve the equation .

Exercise 3

Solve the equation .

Exercise 4

Solve the equation .

Exercise 5

Solve the equation