Intermediate Algebra, Chapter 2, 2.1, Section 2.1, Problem 62

Evaluate the equation $\displaystyle \frac{2x + 5}{5} = \frac{3x + 1}{2} + \frac{-x + 7}{2}$ and check your solution.


$
\begin{equation}
\begin{aligned}

\frac{2x + 5}{5} =& \frac{3x + 1}{2} + \frac{-x + 7}{2}
&& \text{Given equation}
\\
\\
10 \left( \frac{2x + 5}{5} \right) =& 10 \left( \frac{3x + 1}{2} + \frac{-x + 7}{2} \right)
&& \text{Multiply each side by the LCD, } 10
\\
\\
4x + 10 =& 15x + 5 + (-5x) + 35
&& \text{Distributive property}
\\
\\
4x + 10 =& 10x + 40
&& \text{Combine like terms}
\\
\\
4x - 10x =& 40 - 10
&& \text{Subtract $(10x+10)$ from each side}
\\
\\
-6x =& 30
&& \text{Combine like terms}
\\
\\
\frac{-6x}{-6} =& \frac{30}{-6}
&& \text{Divide both sides by $-6$}
\\
\\
x =& -5
&&

\end{aligned}
\end{equation}
$


Checking:


$
\begin{equation}
\begin{aligned}

\frac{2(-5) + 5}{5} =& \frac{3(-5) + 1}{2} + \frac{-(-5) + 7}{2}
&& \text{Let } x = -5
\\
\\
\frac{-10 + 5}{5} =& \frac{-15 + 1}{2} + \frac{5 + 7}{2}
&& \text{Multiply}
\\
\\
\frac{-5}{5} =& \frac{-14}{2} + \frac{12}{2}
&& \text{Add the numerators}
\\
\\
-1 =& -7 + 6
&& \text{Simplify}
\\
\\
-1 =& -1
&& \text{True}

\end{aligned}
\end{equation}
$