Solving Equations

Clearing Fractions and Decimals

Equations are generally easier to solve when they do not contain fractions or decimals. The multiplication principle can be used to“clear” fractions or decimals, as shown here.

In each case, the resulting equation is equivalent to the original equation, but easier to solve.

The easiest way to clear an equation of fractions is to multiply both sides of the equation by the smallest, or least, common denominator.

Example 1

Solve:

a) The number 6 is the least common denominator, so we multiply both sides by 6.

Multiplying both sides by 6

4x - 1 = 12 Simplifying. Note that the fractions are cleared.

-1 = 8x Subtracting 4x from both sides

Dividing both sides by 8

The number checks and is the solution.

b) To solve we can multiply both sides by (or divide by ) to “ undo” the multiplication by on the left side.

Multiplying both sides by

3x + 2 = 20 Simplifying; and

3x = 18 Subtracting 2 from both sides

x = 6 Dividing both sides by 3

The student can confirm that 6 checks and is the solution.

To clear an equation of decimals, we count the greatest number of decimal places in any one number. If the greatest number of decimal places is 1, we multiply both sides by 10; if it is 2, we multiply by 100; and so on.

Example 2

Solve: 16.3 - 7.2y = -8.18

Solution

The greatest number of decimal places in any one number is two. Multiplying by 100 will clear all decimals.

100(16.3 - 7.2y) = 100(-8.18) Multiplying both sides by 100

100(16.3) - 100(7.2y) = 100(-8.18) Using the distributive law

1630 - 720y = -818 Simplifying

- 720y = -818 - 1630 Subtracting 1630 from both sides

- 720y = -2448 Combining like terms

Dividing both sides by -720

y = 3.4

In Example 4, the same solution was found without clearing decimals. Finding the same answer two ways is a good check. The solution is 3.4.

An Equation-Solving Procedure

1. Use the multiplication principle to clear any fractions or decimals. (This is optional, but can ease computations.)
2. If necessary, use the distributive law to remove parentheses. Then combine like terms on each side.
3. Use the addition principle, as needed, to get all variable terms on one side and all constant terms on the other.
4. Combine like terms again, if necessary.
5. Multiply or divide to solve for the variable, using the multiplication principle.
6. Check all possible solutions in the original equation.