To solve the given equation \(\frac{17}{3} - \frac{3}{4} x = \frac{1}{2} x + 5\) step-by-step while incorporating the subtraction and division properties of equality, here’s a detailed breakdown:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Step} & \text{Justification} \\
\hline
\frac{17}{3} - \frac{3}{4} x = \frac{1}{2} x + 5 & \text{Given equation} \\
\hline
\frac{17}{3} - \frac{3}{4} x - \frac{17}{3} = \frac{1}{2} x + 5 - \frac{17}{3} & \text{Subtraction property of equality: Subtract } \frac{17}{3} \text{ from both sides} \\
\hline
-\frac{3}{4} x = \frac{1}{2} x + 5 - \frac{17}{3} & \text{Simplification: cancel } \frac{17}{3} - \frac{17}{3} \text{ on the left} \\
\hline
-\frac{3}{4} x = \frac{1}{2} x - \frac{2}{3} & \text{Simplification: convert 5 into a common denominator and combine terms on the right} \\
\hline
-\frac{3}{4} x - \frac{1}{2} x = \frac{1}{2} x - \frac{2}{3} - \frac{1}{2} x & \text{Subtraction property of equality: Subtract } \frac{1}{2} x \text{ from both sides} \\
\hline
-\frac{5}{4} x = -\frac{2}{3} & \text{Combine like terms on both sides} \\
\hline
x = \frac{-\frac{2}{3} \cdot -\frac{4}{5}}{-\frac{5}{4}\cdot\frac{4}{5} } &
\text{Division property of equality: Multiply both sides by } -\frac{4}{5}\\
\hline
x =\frac {-2 \cdot -4}{15} \cdot \frac{-5 }{-20} & \\
\hline
x = 0.5333333333333333 = \frac{8}{15} & \text{Simplified final result} \\
\hline
\end{array}
\][/tex]
This solution verifies each step methodically from the initial equation to the simplified solution, ensuring that all properties of equality and algebraic manipulations are properly applied.
