\begin{tabular}{|c|l|}
\hline Step & Justification \\
\hline [tex]$\frac{17}{3}-\frac{3}{4} x=\frac{1}{2} x+5$[/tex] & \\
\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 \\
\hline [tex]$-\frac{3}{4} x=\frac{1}{2} x-\frac{2}{3}$[/tex] & Simplification \\
\hline [tex]$-\frac{3}{4} x-\frac{1}{2} x=\frac{1}{2} x-\frac{2}{3}-\frac{1}{2} x$[/tex] & Subtraction property of equality \\
\hline [tex]$-\frac{5}{4} x=-\frac{2}{3}$[/tex] & Simplification \\
\hline [tex]$-\frac{5}{4} x \cdot -\frac{4}{5}=-\frac{2}{3} \cdot -\frac{4}{5}$[/tex] & Multiplication property of equality \\
\hline [tex]$x=\frac{8}{15}$[/tex] & Simplification \\
\hline
\end{tabular}



Answer :

Let's solve the equation step-by-step while applying the appropriate mathematical properties. The original equation is:

[tex]\[ \frac{17}{3} - \frac{3}{4}x = \frac{1}{2}x + 5 \][/tex]

Step 1: Start with the given equation.

[tex]\[ \frac{17}{3} - \frac{3}{4}x = \frac{1}{2}x + 5 \][/tex]
Justification: The original equation given to us.

Step 2: Apply the Subtraction Property of Equality to subtract [tex]\(\frac{17}{3}\)[/tex] from both sides:

[tex]\[ \frac{17}{3} - \frac{3}{4}x - \frac{17}{3} = \frac{1}{2}x + 5 - \frac{17}{3} \][/tex]
Justification: Subtract the same value ([tex]\(\frac{17}{3}\)[/tex]) from both sides to keep the equation balanced.

Step 3: Simplify both sides:

[tex]\[ -\frac{3}{4}x = \frac{1}{2}x + 5 - \frac{17}{3} \][/tex]
Justification: Simplify the left-hand side by cancelling out [tex]\(\frac{17}{3}\)[/tex].

Next, convert the constant on the right-hand side to a common denominator:

[tex]\[ 5 - \frac{17}{3} = \frac{15}{3} - \frac{17}{3} = -\frac{2}{3} \][/tex]
So,

[tex]\[ -\frac{3}{4}x = \frac{1}{2}x - \frac{2}{3} \][/tex]
Justification: Simplify the arithmetic to combine terms on the right-hand side.

Step 4: Apply the Subtraction Property of Equality to subtract [tex]\(\frac{1}{2}x\)[/tex] from both sides:

[tex]\[ -\frac{3}{4}x - \frac{1}{2}x = \frac{1}{2}x - \frac{2}{3} - \frac{1}{2}x \][/tex]
Justification: Subtract [tex]\(\frac{1}{2}x\)[/tex] from both sides to isolate [tex]\(x\)[/tex] terms on the left-hand side.

Step 5: Simplify both sides:

[tex]\[ -\frac{3}{4}x - \frac{1}{2}x = -\frac{2}{3} \][/tex]
Combine like terms on the left-hand side.

To do this, find a common denominator for the coefficients of [tex]\(x\)[/tex]:

[tex]\[ -\frac{3}{4}x - \frac{1}{2}x = -\frac{3}{4}x - \frac{2}{4}x = -\frac{5}{4}x \][/tex]

So,

[tex]\[ -\frac{5}{4}x = -\frac{2}{3} \][/tex]

Step 6: Apply the Multiplication Property of Equality to isolate [tex]\(x\)[/tex]. Multiply both sides by [tex]\(-\frac{4}{5}\)[/tex]:

[tex]\[ -\frac{5}{4}x \cdot -\frac{4}{5} = -\frac{2}{3} \cdot -\frac{4}{5} \][/tex]
Justification: To solve for [tex]\(x\)[/tex], multiply both sides by the reciprocal of [tex]\(-\frac{5}{4}\)[/tex], which is [tex]\(-\frac{4}{5}\)[/tex].

Step 7: Simplify both sides:

[tex]\[ x = \frac{8}{15} \][/tex]
Justification: Simplify the right side [tex]\((-\frac{2}{3} \cdot -\frac{4}{5} = \frac{8}{15})\)[/tex].

Therefore, the solution to the equation is:

[tex]\[ x = \frac{8}{15} \][/tex]

Each step aligns with a mathematical principle to ensure the correctness of the solution.