Let's write out the correct solution for the given equation step-by-step, including the appropriate justification for each step.
### Step-by-step solution:
1. Step 1: Original equation
- Statements: [tex]$-6 = -\frac{2}{3}(x + 12) + \frac{1}{3} x$[/tex]
- Reasons: given
[tex]\[
\begin{array}{|c|c|c|}
\hline \text{Step} & \text{Statements} & \text{Reasons} \\
\hline 1 & -6 = -\frac{2}{3}(x + 12) + \frac{1}{3} x & \text{given} \\
\hline
\end{array}
\][/tex]
2. Step 2: Distribute [tex]\(-\frac{2}{3}\)[/tex] through [tex]\((x + 12)\)[/tex]
- Statements: [tex]$-6 = -\frac{2}{3} x - 8 + \frac{1}{3} x$[/tex]
- Reasons: distribute [tex]\(-\frac{2}{3}\)[/tex]
[tex]\[
\begin{array}{|c|c|c|}
\hline 2 & -6 = -\frac{2}{3} x - 8 + \frac{1}{3} x & \text{distribute \(-\frac{2}{3}\)} \\
\hline
\end{array}
\][/tex]
3. Step 3: Combine like terms
- Statements: [tex]$-6 = -8 - \frac{1}{3} x$[/tex]
- Reasons: combine like terms
[tex]\[
\begin{array}{|c|c|c|}
\hline 3 & -6 = -8 - \frac{1}{3} x & \text{combine like terms} \\
\hline
\end{array}
\][/tex]
4. Step 4: Add 8 to both sides to isolate the term with [tex]\(x\)[/tex]
- Statements: [tex]$2 = -\frac{1}{3} x$[/tex]
- Reasons: add 8 to both sides
[tex]\[
\begin{array}{|c|c|c|}
\hline 4 & 2 = -\frac{1}{3} x & \text{add 8 to both sides} \\
\hline
\end{array}
\][/tex]
5. Step 5: Multiply both sides by [tex]\(-3\)[/tex] to solve for [tex]\(x\)[/tex]
- Statements: [tex]$x = -6$[/tex]
- Reasons: multiply by [tex]\(-3\)[/tex]
[tex]\[
\begin{array}{|c|c|c|}
\hline 5 & x = -6 & \text{multiply by \(-3\)} \\
\hline
\end{array}
\][/tex]
### Summary:
The complete steps with their justifications are as follows:
[tex]\[
\begin{array}{|c|c|c|}
\hline \text{Step} & \text{Statements} & \text{Reasons} \\
\hline 1 & -6 = -\frac{2}{3}(x + 12) + \frac{1}{3} x & \text{given} \\
\hline 2 & -6 = -\frac{2}{3} x - 8 + \frac{1}{3} x & \text{distribute \(-\frac{2}{3}\)} \\
\hline 3 & -6 = -8 - \frac{1}{3} x & \text{combine like terms} \\
\hline 4 & 2 = -\frac{1}{3} x & \text{add 8 to both sides} \\
\hline 5 & x = -6 & \text{multiply by \(-3\)} \\
\hline
\end{array}
\][/tex]
