###### Example3.5.4

Solve $$\frac{x}{8} = \frac{15}{12}$$ for $$x\text{.}$$

Instead of finding the LCD of the two fractions, we'll simply multiply both sides of the equation by $$8$$ and by $$12\text{.}$$ This will still have the effect of canceling the denominators on both sides of the equation.

\begin{align*} \frac{x}{8} \amp= \frac{15}{12}\\ \highlight{12\cdot8\cdot}\frac{x}{8} \amp= \frac{15}{12}\highlight{\cdot12\cdot8}\\ 12\cdot\cancelhighlight{8}\cdot\frac{x}{\cancelhighlight{8}} \amp= \frac{15}{\cancelhighlight{12}}\cdot\cancelhighlight{12}\cdot8\\ 12\cdot x \amp= 15\cdot 8\\ 12x \amp= 120 \\ \divideunder{12x}{12} \amp= \divideunder{120}{12} \\ x \amp= 10 \end{align*}

Our work indicates $$10$$ is the solution. We can check this as we would for any equation, by substituting $$10$$ for $$x$$ and verifying we obtain a true statement:

\begin{align*} \frac{10}{8} \amp\stackrel{?}{=} \frac{15}{12}\\ \frac{5}{4} \amp\stackrel{\checkmark}{=} \frac{5}{4} \end{align*}

Since both fractions reduce to $$\frac{5}{4}\text{,}$$ we know the solution to the equation $$\frac{x}{8} = \frac{15}{12}$$ is $$10$$ and the solution set is $$\left\{10\right\}\text{.}$$

in-context