Chapter 12.1, Problem 34SR

To determine

To solve: The given equation and state any extraneous solutions.

Answer to Problem 34SR

The solution of the equation is $\frac{9}{2}$ .

Explanation of Solution

Given information:

The following equation:

  $\frac{6}{5}+\frac{4p}{3}=\frac{8p}{5}$

Calculation:

The equation is solved as follows:

  $\begin{array}{l}\frac{6}{5}+\frac{4p}{3}=\frac{8p}{5} [\text{Original equation]}\\ \frac{3(6)+5(4p)}{15}=\frac{8p}{5} [\text{Add fractions]}\\ \frac{18+20p}{15}=\frac{8p}{5} [\text{Simplify]}\\ \text{ 5}(18+20p)=15(8p)\text{ [Form cross product]}\\ \text{ 90}+100p=\text{ 120}p [\text{Distributive property]}\\ \underset{\_}{-120p\text{ =}-120p} [\text{Subtract 120}p\text{ from both sides]}\\ 90-20p=\text{ 0 } [\text{Simplify]}\\ \underset{\_}{-90=-90} [\text{Subtract 90 from both sides]}\\ -20p=-90 [\text{Simplify]}\\ \frac{-20p}{-20}=\frac{-90}{-20} [\text{Divide both sides by }-2\text{0]}\\ p=\frac{9}{2}\text{ [Simplify]}\end{array}$

To check the solution, substitute $\frac{9}{2}$ for $p$ in the original equation

  $\begin{array}{l}\frac{6}{5}+\frac{4p}{3}=\frac{8p}{5} [\text{Original equation}]\\ \frac{6}{5}+\frac{4\left(\frac{9}{2}\right)}{3}=\frac{8\left(\frac{9}{2}\right)}{5}\left[\text{Substitute }\frac{9}{2}\text{ for }p\right]\\ \frac{6}{5}+\frac{18}{3}=\frac{36}{5}\text{ [Simplify]}\\ \frac{18+90}{15}=\frac{36}{5} [\text{Add]}\\ \frac{108}{15}=\frac{36}{5}\text{ [Simplify]}\\ \frac{36}{5}\text{=}\frac{36}{5} [\text{Divivde common factors]}\end{array}$

Since both sides are equal, the solution is correct.