Question

Solve each equation. State any extraneous solutions.

$\frac{\mathbf{3}\mathbf{m}}{\mathbf{4}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{=}\frac{\mathbf{3}\mathbf{m}\mathbf{+}\mathbf{4}}{\mathbf{6}}$

Short answer

The solution of the equation is$\frac{\mathbf{4}}{\mathbf{3}}$

Step-by-step solution

Step 1. State the concept for observational studies.

The linear equations in one variable is an equation which is expressed in the form of $ax+b=0$, where a and b are two integers, and x is a variable and has only one solution.

Step 2. Solve the equation.

The equation can be solved as follows:

$\begin{array}{c}\frac{3m}{4}+\frac{1}{3}=\frac{3m+4}{6}\\ \frac{3\left(3m\right)+4\left(1\right)}{12}=\frac{3m+4}{6}\\ \frac{9m+4}{12}=\frac{3m+4}{6}\\ \frac{9m+4}{12}\times 12=\frac{3m+4}{6}\times 12\\ 9m+4=\left(3m+4\right)\times 2\\ 9m+4=6m+8\\ 9m-6m=8-4\\ 6m=4\\ m=\frac{4}{3}\end{array}$

Step 3. Check the solution.

To check the solution, substitute $m=\frac{4}{3}$for in the original equation,

$\begin{array}{c}\frac{3m}{4}+\frac{1}{3}=\frac{3m+4}{6}\\ \frac{3\left(\frac{4}{3}\right)}{4}+\frac{1}{3}=\frac{3\left(\frac{4}{3}\right)+4}{6}\\ \frac{4}{4}+\frac{1}{3}=\frac{4+4}{6}\\ 1+\frac{1}{3}=\frac{8}{6}\\ \frac{3+1}{3}=\frac{4}{3}\\ \frac{4}{3}=\frac{4}{3}\\ LHS=RHS\end{array}$

Since both sides are equal, the equation is correct.