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Let's solve this step-by-step and match the appropriate property with each equation transformation:

1. **Step 1**: \( 4(x + 7) = 9x - 7 \)
   - **Reason**: This is the **original equation** (already provided).

2. **Step 2**: \( 4x + 28 = 9x - 7 \)
   - **Reason**: This transformation uses the **distributive property** (distributing \( 4 \) to both \( x \) and \( 7 \)).

3. **Step 3**: \( 28 = 5x - 7 \)
   - **Reason**: Here, the \( 4x \) is subtracted from both sides, leaving \( 28 = 5x - 7 \). This applies the **subtraction property of equality**.

4. **Step 4**: \( 35 = 5x \)
   - **Reason**: Adding \( 7 \) to both sides eliminates the \( -7 \), using the **addition property of equality**.

5. **Step 5**: \( 7 = x \)
   - **Reason**: Dividing both sides by \( 5 \) gives \( x \), which uses the **division property of equality**.

Thus, the table would look like this:

| Step | Equation               | Reason                          |
|------|------------------------|---------------------------------|
| 1    | \( 4(x + 7) = 9x - 7 \) | original equation               |
| 2    | \( 4x + 28 = 9x - 7 \)  | distributive property           |
| 3    | \( 28 = 5x - 7 \)       | subtraction property of equality|
| 4    | \( 35 = 5x \)           | addition property of equality   |
| 5    | \( 7 = x \)             | division property of equality   |

Would you like further details or have any questions about this process?

Here are some related questions for you to explore:

1. What is the associative property and when would you use it in solving equations?
2. How does the commutative property differ from the distributive property?
3. Why is it important to understand properties of equality in solving equations?
4. Can you provide an example where the multiplication property of equality is used?
5. What is the reflexive property of equality and how might it apply to algebra?

**Tip**: The distributive property allows you to multiply a number by a group of terms inside parentheses, a key step in simplifying algebraic expressions.

The steps to correctly solve an equation are shown. Move a property into each box of the Reason column to justify each step.

Learn how to solve a linear equation step-by-step and justify each transformation using properties of equality. This problem involves key algebraic concepts like the distributive property and addition/subtraction properties of equality, suitable for Grades 8-10.

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