Let's complete the missing reasons for the given proof step-by-step, ensuring we thoroughly understand each mathematical property applied.
Given equation: [tex]\( 4(x-2) = 6x + 18 \)[/tex]
[tex]\[
\begin{tabular}{|c|c|}
\hline
\text{Statements} & \text{Reasons} \\
\hline
1. \; 4(x-2) = 6x + 18 & \text{given} \\
\hline
2. \; 4x - 8 = 6x + 18 & \text{distributive property} \\
\hline
3. \; -2x - 8 = 18 & \text{subtraction property of equality} \\
\hline
4. \; -2x = 26 & \text{addition property of equality} \\
\hline
5. \; x = -13 & \text{division property of equality} \\
\hline
\end{tabular}
\][/tex]
### Detailed Reasoning:
1. Given Equation:
- Statement 1: [tex]\(4(x - 2) = 6x + 18\)[/tex]
- Reason: This is the given equation.
2. Distributive Property:
- Statement 2: [tex]\(4(x - 2) = 6x + 18\)[/tex] simplifies to [tex]\(4x - 8 = 6x + 18\)[/tex].
- Reason: By applying the distributive property [tex]\(a(b + c) = ab + ac\)[/tex], we get [tex]\(4 \cdot x - 4 \cdot 2\)[/tex], which simplifies to [tex]\(4x - 8\)[/tex].
3. Subtraction Property of Equality:
- Statement 3: [tex]\(4x - 8 = 6x + 18\)[/tex] becomes [tex]\(-2x - 8 = 18\)[/tex].
- Reason: Subtract [tex]\(6x\)[/tex] from both sides to isolate the variable term on one side: [tex]\(4x - 8 - 6x = 6x + 18 - 6x\)[/tex], which simplifies to [tex]\(-2x - 8 = 18\)[/tex].
4. Addition Property of Equality:
- Statement 4: [tex]\(-2x - 8 = 18\)[/tex] simplifies to [tex]\(-2x = 26\)[/tex].
- Reason: Add 8 to both sides to isolate the term with the variable: [tex]\(-2x - 8 + 8 = 18 + 8\)[/tex], which simplifies to [tex]\(-2x = 26\)[/tex].
5. Division Property of Equality:
- Statement 5: [tex]\(-2x = 26\)[/tex] simplifies to [tex]\(x = -13\)[/tex].
- Reason: Divide both sides by -2 to solve for [tex]\(x\)[/tex]: [tex]\(\frac{-2x}{-2} = \frac{26}{-2}\)[/tex], which simplifies to [tex]\(x = -13\)[/tex].
Thus, the completed table with the correct reasons is:
[tex]\[
\begin{tabular}{|c|c|}
\hline
\text{Statements} & \text{Reasons} \\
\hline
1. \; 4(x-2) = 6x + 18 & \text{given} \\
\hline
2. \; 4x - 8 = 6x + 18 & \text{distributive property} \\
\hline
3. \; -2x - 8 = 18 & \text{subtraction property of equality} \\
\hline
4. \; -2x = 26 & \text{addition property of equality} \\
\hline
5. \; x = -13 & \text{division property of equality} \\
\hline
\end{tabular}
\][/tex]
The missing reasons are:
- Step 3: Subtraction property of equality
- Step 5: Division property of equality
Hence, the correct answer is:
3. subtraction property of equality; 5. division property of equality.
