Eliminating Fractions

Linear Equations Containing Fractions

When a fraction is multiplied by a number which divides evenly by the denominator, the result is an integer.

For example,  can be simplified as:

In other words, we simply divide 20 by 5, and then multiply what's left.

Manipulating fractions in an equation can be tedious.  Fortunately it is easy to transform an equation with fractions into a simpler equation without fractions.  The method is to multiply both sides by the least common denominator of the fractions, use the distributive property, and then cancel the fractions.

We will illustrate this idea through examples.

Example 1:  Solve the following equation by first eliminating fractions.

Solution:  The least common denominator is 6, so we multiply both sides by 6 and then simplify:

Now subtract 2x from both sides:

So x = 15 is the solution.

Example 2:  Solve this equation:

Solution:  The least common denominator is 12:

Now cancel the denominators, but be sure to enclose the numerators in parentheses:

Subtract 5x and then divide by 5:

So x = 1 is the solution.

Example 3:  Solve this equation:

Solution:  Multiply both sides by 8:

Now subtract 8x:

Divide by 7 to get

Nonlinear equations that can be made linear

An equation with x in the denominator of a fraction is not a linear equation, but some such equations can be transformed into linear equations by the above method.

Example 4:  Solve this equation:

Solution:  The least common denominator is 3x:

Example 5:  Solve:

Solution:  The least common denominator is 6(x – 2):