## Thinking Mathematically (6th Edition)

By the provided steps , one obtains the correct solution when multiply both sides of the equation$\frac{x}{5}-\frac{x}{2}=1$ by 20 but still it is preferable to multiply such type of equation with least common denominator 10, since least common denominator is the smallest common number that can divide all denominators and eliminate the fractions of the equation in much simplified manner.
Consider an equation, $\frac{x}{5}-\frac{x}{2}=1$. Multiply both sides of equation$\frac{x}{5}-\frac{x}{2}=1$by 20; $20\left( \frac{x}{5}-\frac{x}{2} \right)=20\left( 1 \right)$ Use distributive property to simplify the equation, $4x-10x=20$. Solve it further for x; $-6x=20$ Divide both sides by$-6$; \begin{align} & \left( -\frac{1}{6} \right)\left( -6x \right)=\left( -\frac{1}{6} \right)\left( 20 \right) \\ & x=-\frac{10}{3} \end{align} Thus, solution for equation$\frac{x}{5}-\frac{x}{2}=1$ is$x=-\frac{10}{3}$. Now, verify if this is the right solution for the provided equation. For this substitute this value of $x=-\frac{10}{3}$in the original equation. \begin{align} & \frac{x}{5}-\frac{x}{2}=1 \\ & \left( \frac{1}{5} \right)\left( -\frac{10}{3} \right)-\left( \frac{1}{2} \right)\left( -\frac{10}{3} \right)=1 \\ & -\frac{2}{3}+\frac{5}{3}=1 \\ & \frac{-2+5}{3}=1 \end{align} So, \begin{align} & \frac{3}{3}=1 \\ & 1=1 \end{align} This means that the solution for equation$\frac{x}{5}-\frac{x}{2}=1$ is$x=-\frac{10}{3}$, is the correct solution. Now, multiply both sides of the same equation x by least common denominator 10 and check whether it provides the same solution as above. Divide both sides by ; Yes, it is the same solution but also notice that calculation is more simplified when multiply this equation with least common denominator 10, since least common denominator is the smallest common number that can divide all denominators and eliminate the fractions of the equation..