Chapter 11, Problem 24PT

To determine

Find the solution of $\frac{m}{m-3}+\frac{1}{4}=\frac{3}{m-3}$ and explain the error of your friend.

Answer to Problem 24PT

No solution

Explanation of Solution

Given:

Your friend says the solution of the rational equation $\frac{m}{m-3}+\frac{1}{4}=\frac{3}{m-3}$ is 3. Explain the error that your friend may have made.

Concept Used:

To get rid from the denominators, multiply each term by the LCM of the denominators. The LCM of the denominators is $4(m-3)$

Calculation:

To get rid from the denominators, multiply each term by the LCM of the denominators. The LCM of the denominators is $4(m-3)$

  $\begin{array}{l}\frac{m}{m-3}+\frac{1}{4}=\frac{3}{m-3}\\ 4(m-3)·\frac{m}{m-3}+4(m-3)·\frac{1}{4}=4(m-3)·\frac{3}{m-3}[\text{Multiply each side by }4(m-3)]\\ 4m+m-3=12\Rightarrow 5m-3=12\\ 5m-3+3=12+3\\ 5m=15\Rightarrow \frac{5m}{5}=\frac{15}{5}\Rightarrow m=3\end{array}$

Rational expression is in the form $\frac{\text{P}}{\text{Q}}$ , where $\text{Q}\ne 0$ , denominator cannot be zero in any rational expression.

Excluded values are also called points of discontinuity. These are the values that make the denominator equal to zero and are not part of the domain.

  $m=3$ is the excluded value of the equation $\frac{m}{m-3}+\frac{1}{4}=\frac{3}{m-3}$

There is no solution for this equation $\frac{m}{m-3}+\frac{1}{4}=\frac{3}{m-3}$ .

You friend has not noticed the excluded or undefined value $m=3$ .

Thus, No solution