Question

Explain how you know which method you should use to solve a rational equation.

Answer 1

Methods to solve rational equation:

Rational equation:

A rational equation is an equation containing at least one rational expression.

Method 1:

The method for solving rational equations is to rewrite the rational expressions in terms of a common denominator. Then, since we know the numerators are equal, we can solve for the variable.

For example,

`(1)/(2)=(x)/(2)\Rightarrow x=1`

This can be used for rational equations with polynomials too.

For example,

`(1+x)/(x-3)=(4)/(x-3)\Rightarrow(1+x)=4 \Rightarrow x=3`

When the terms in a rational equation have unlike denominators, solving the equation will be as follows

`(x+2)/(8)=(3)/(4)`

`\Rightarrow(x+2)/(1)=(3*8)/(4)\Rightarrow{x+2}=2*3`

`\Rightarrow{x+2}=6\Rightarrow x=6-2\Rightarrow x=4`

Method 2:

Another way of solving the above equation is by finding least common denominator (LCD)

`(x+2)/(8)=(3)/(4)`

Factors of 4:
`1*2*2`

Factors of 8:
`1*2*2*2`

The LCD of 4 and 8 is 8. So, we have to make the right hand side denominator as 8. This is done by the following step,

`\Rightarrow(x+2)/(8)=(3)/(4)*{2}{2}`

we get,

`\Rightarrow(x+2)/(8)=(6)/(8)`

On cancelling 8 on both sides we get,

`\Rightarrow(x+2)=6\rightarrow x=6-2\rightarrow x=4`

Hence, these are the ways to solve a rational equation.