How to solve an equation that has a variable in the denominator?

The equations that have variables in the denominator are termed as rational equations. Even though this may be a new name for you, it uses the same concepts used in solving regular algebra equations.

Here are some e xamples of rational equations:

To solve rational equations, you should first convert them into a standard linear or quadratic form. This can be done using one or more of the following steps. Usually the simpler rational equations may not involve all of these steps:

1. Isolate the terms with variable on one side and constants on the other side
2. Add or subtract terms with variable
3. Cross-multiply both sides to get a standard linear or quadratic equation

Let’s take an example involving a simple rational equation that converts to a linear form:

Example:

What is the value of   if ?

A) 1

B) 2

C) 3

D) 4

E) 5

Solution:

Step 1: Here we see that all variable terms are already on one side. Therefore we first use the cross-multiplication to simplify the equation:

Step 2: Next we get rid of the parentheses:

Step 3: Finally isolate the variable to solve the equation

D is the correct answer.

Sometimes you can also simplify the equation into a linear or quadratic form a lot quicker by multiplying both sides of the equation with the denominator containing the variable term. Let’s look at a grid-in example:

Example:

If   what is a possible value of ?

Solution:

This question can be easily solved by plugging in the values if you have the luxury of answer choices. But alas! This seems to be a fill in the blank type of question. Sadly you will have to solve this question using math concepts.

Step 1: To solve the question we need to eliminate the  in the denominator. This can be done by multiplying both sides of the equation by 

Step 2: The next step is to factorize the quadratic equation:

Either 1 or 2 could be the correct answer.