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# Solving Linear Equations Containing Fractions

To solve a linear equation that contains fractions, it is often helpful to first clear the fractions. Then the equation can be solved using the steps for solving a linear equation.

One way to clear the fractions is to multiply both sides of the equation by the least common denominator (LCD) of the fractions. The LCD is the smallest positive number that is evenly divisible by all of the denominators in the equation.

Here is a procedure for finding the LCD of a set of fractions.

Procedure

To Find the Least Common Denominator (LCD)

Step 1 Factor each denominator into its prime factors.

Step 2 List each factor the greatest number of times that it appears in any one of the denominators.

Step 3 Multiply the prime factors in the list.

Example 1

Find the LCD of Solution

 Solution Step 1 Factor each denominator into its prime factors. 10 = 2 Â· 5 12 = 2 Â· 2 Â· 3 15 = 3 Â· 5 Step 2 List each factor the greatest number of times that it appears in any one of the denominators. Step 3 Multiply the prime factors in the list. 2, 2, 3, 52 Â· 2 Â· 3 Â· 5 = 60

The LCD of That is, 60 is the smallest number that is evenly divisible by 10, 12, and 15.

Now, letâ€™s solve an equation that contains fractions.

Example 2

Solve: Solution

First, find the LCD of the fractions. The LCD is the smallest number that is evenly divisible by 6 and 9. The LCD is 18.

Next, clear the fractions:

 â€¢ Multiply each side by the LCD, 18.  â€¢ Simplify. Then, solve the equation for x: â€¢ Remove the parentheses. â€¢ Subtract 2x from both sides. We leave the check to you.  The solution is x = 36. 3x  3x x = 2(x + 18)  = 2x + 36 = 36
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