Classwork worksheet on properties of equality with algebraic equations and solution steps.

To solve the problem, we need to identify the property of equality that justifies each step in the given equations. Let's go through each table step by step.

Table 1:

Equation: \( x + (x - 6) = 8 \)

| Equation | Steps |
|-------------------|---------------------------------|
| \( x + (x - 6) = 8 \) | Original Equation |
| \( 2x - 6 = 8 \) | Combining Like Terms |
| | Addition Property of Equality |
| \( x = 7 \) | Division Property of Equality |

- Step 1: \( x + (x - 6) = 8 \) → \( 2x - 6 = 8 \)
- Justification: Combining like terms (simplifying the left side).

- Step 2: \( 2x - 6 = 8 \) → \( 2x = 14 \)
- Justification: Addition Property of Equality (add 6 to both sides).

- Step 3: \( 2x = 14 \) → \( x = 7 \)
- Justification: Division Property of Equality (divide both sides by 2).

Table 2:

Equation: \( x + (4x + 12) = 12 \)

| Equation | Steps |
|-------------------|---------------------------------|
| \( x + (4x + 12) = 12 \) | Original Equation |
| \( 5x + 12 = 22 \) | Combining Like Terms |
| \( 5x = 10 \) | Subtraction Property of Equality |
| | Division Property of Equality |
| \( x = 2 \) | |

- Step 1: \( x + (4x + 12) = 12 \) → \( 5x + 12 = 22 \)
- Justification: Combining like terms (simplifying the left side).

- Step 2: \( 5x + 12 = 22 \) → \( 5x = 10 \)
- Justification: Subtraction Property of Equality (subtract 12 from both sides).

- Step 3: \( 5x = 10 \) → \( x = 2 \)
- Justification: Division Property of Equality (divide both sides by 5).

Table 3:

Equation: \( 4(x - 6) = 40 \)

| Equation | Steps |
|-------------------|---------------------------------|
| \( 4(x - 6) = 40 \) | Original Equation |
| \( x - 6 = 10 \) | Division Property of Equality |
| \( x = 16 \) | Addition Property of Equality |

- Step 1: \( 4(x - 6) = 40 \) → \( x - 6 = 10 \)
- Justification: Division Property of Equality (divide both sides by 4).

- Step 2: \( x - 6 = 10 \) → \( x = 16 \)
- Justification: Addition Property of Equality (add 6 to both sides).

Table 4:

Equation: \( 5x + (3(x + 4)) = 28 \)

| Equation | Steps |
|-------------------|---------------------------------|
| \( 5x + (3(x + 4)) = 28 \) | Original Equation |
| \( 5x + 3x + 12 = 28 \) | Distributive Property |
| \( 8x + 12 = 28 \) | Combining Like Terms |
| | Subtraction Property of Equality |
| \( x = 2 \) | Division Property of Equality |

- Step 1: \( 5x + (3(x + 4)) = 28 \) → \( 5x + 3x + 12 = 28 \)
- Justification: Distributive Property (distribute 3 across \( x + 4 \)).

- Step 2: \( 5x + 3x + 12 = 28 \) → \( 8x + 12 = 28 \)
- Justification: Combining like terms (simplify the left side).

- Step 3: \( 8x + 12 = 28 \) → \( 8x = 16 \)
- Justification: Subtraction Property of Equality (subtract 12 from both sides).

- Step 4: \( 8x = 16 \) → \( x = 2 \)
- Justification: Division Property of Equality (divide both sides by 8).

Final Answer:

\[
\boxed{
\begin{array}{c|c}
\text{Table 1} & \text{Table 2} \\
\hline
\text{Combining Like Terms} & \text{Combining Like Terms} \\
\text{Addition Property of Equality} & \text{Subtraction Property of Equality} \\
\text{Division Property of Equality} & \text{Division Property of Equality} \\
\hline
\text{Table 3} & \text{Table 4} \\
\hline
\text{Division Property of Equality} & \text{Distributive Property} \\
\text{Addition Property of Equality} & \text{Combining Like Terms} \\
& \text{Subtraction Property of Equality} \\
& \text{Division Property of Equality} \\
\end{array}
}
\]