Solving Fractional Equations
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Solving fractional equations is very much like addition and subtraction of fractions, but after the first step, you will GET RID of the denominators!  "Getting rid" of the denominators is actually changing them all to the value of 1, which of course does not need to be written, leaving us with a denominator-free equation.  YEA!!

To "get rid" of the denominators,

What to do..... Why it works ...
1.  start by choosing the common denominator for the equation. Terms in equations are often connected by addition or subtraction.  Dealing with addition or subtraction of fractions requires a common denominator.
2.  multiply EVERY TERM in the equation by the common denominator.  In an equation, (unlike an expression), you may multiply "every term" on both sides of the equal sign by the same value and not change the equation.  You maintain a balanced equation. 
3.  reduce each term to form a "denominator free" equation.  All denominators in the equation can be reduced with our common denominator, thus leaving all equation denominators as a value of one. 

Example 1:

Solve:
 

Since 5 is the only denominator, multiply the entire equation by 5:

So:

becomes

which is

2x + 5 = 13 and  x = 4
Remember to multiply every term by 5, including the 1 (from the equation).

Answer:  x = 4

 

   

Example 2:

Solve:

The common denominator is 15.  Multiply every term by 15:

5(x) 3(2x) = - 7
5x 6x = - 7
 -x = -7, so x = 7

Answer:  x = 7

 

Example 3:

Solve:  

The common denominator is 10.  Multiply every term by 10:

Answer:  x = -64

In this problem, since the first numerator (x + 3) contained more than one term, it was necessary to use parentheses when multiplying.  For problems like this one, be sure to WRITE OUT the problem using ( ) to avoid making careless errors.

 

Example 4:

Solve:  

The common denominator is 2x.  Multiply each term by 2x:

Answer:  x = 5 and x = -2

Since this problem is in the form of a proportion, it can also be solved by using "cross multiply".  (In a proportion, the product of the means equals the product of the extremes.)


 

See how to use your
TI-83+/84+ graphing calculator  for solving equations.
Click calculator.