To solve the equation [tex]\(\frac{17}{3} - \frac{3}{4} x = \frac{1}{2} x + 5\)[/tex] step-by-step, we will use properties of equality and simplification rules. Here is the detailed solution:
\begin{tabular}{|c|l|}
\hline
Step & Justification \\
\hline
[tex]\(\frac{17}{3} - \frac{3}{4} x = \frac{1}{2} x + 5\)[/tex] & Given equation \\
\hline
[tex]\(\frac{17}{3} - \frac{3}{4} x - \frac{17}{3} = \frac{1}{2} x + 5 - \frac{17}{3}\)[/tex] & Subtraction property of equality: Subtract [tex]\(\frac{17}{3}\)[/tex] from both sides \\
\hline
[tex]\(- \frac{3}{4} x = \frac{1}{2} x - \frac{2}{3}\)[/tex] & Simplification: [tex]\(\frac{17}{3} - \frac{17}{3} = 0\)[/tex] on the left-hand side and combine the constants on the right-hand side \\
\hline
[tex]\(-\frac{3}{4} x - \frac{1}{2} x = \frac{1}{2} x - \frac{2}{3} - \frac{1}{2} x\)[/tex] & Addition/Subtraction property of equality: Subtract [tex]\(\frac{1}{2} x\)[/tex] from both sides \\
\hline
[tex]\(-\frac{5}{4} x = -\frac{2}{3}\)[/tex] & Simplification: Combine like terms on both sides ([tex]\(-\frac{3}{4} x - \frac{1}{2} x\)[/tex] becomes [tex]\(-\frac{5}{4} x\)[/tex]) \\
\hline
\end{tabular}
Thus, the simplified form of the given equation after performing the necessary steps is [tex]\(-\frac{5}{4} x = -\frac{2}{3}\)[/tex].
