Question

Solve the equation justifying each step with the correct reasoning.

2(x + 8) = 2x + 8

Step 1: _______ Property to get Response area simplified equation

Step 2: _______ Property to get Response area simplified equation

For this equation, there is/are ____________

properties and reasons:
1. subtraction of equality
2. Multiplication of equality
3. combine like terms
4. division of equality
5. addition of equality
6. distributive

Equation simplified:
1. One solution
2. 16 = 8
3. 2x + 16 = 2x + 8
4. 2x + 8 = 2x + 8
5. No solution
6. x = -8
7. x + 16 = 8
8. 2x + 8 = 8
9. x = 2
10. x + 8 = 8

Answer 1

I can only use each option only once

Answer 2

To solve the equation \(2(x + 8) = 2x + 8\) and justify each step, we can proceed as follows:

**Step 1:** Use the **Distributive Property** to simplify the left side of the equation.

\[
2(x + 8) = 2x + 16
\]

Now the equation reads:

\[
2x + 16 = 2x + 8
\]

**Step 2:** Use the **Subtraction of Equality** property to isolate the variable.

Subtract \(2x\) from both sides:

\[
2x + 16 - 2x = 2x + 8 - 2x
\]

This simplifies to:

\[
16 = 8
\]

At this point, we observe that we have an equation that is not true (16 does not equal 8). This indicates that there are no solutions to the original equation.

For this equation, there are:

1. Properties used:
- **Distributive Property** (used in Step 1)
- **Subtraction of Equality** (used in Step 2)

Thus, the final conclusion is:

**There is no solution.**

The clearly stated steps would be:
- Step 1: **Distributive Property**
- Step 2: **Subtraction of Equality**

And the final answer for the nature of solutions:
- There are **no solutions** because the equation \(16 = 8\) is false.

Answer 3

Got it! Let's solve the equation \(2(x + 8) = 2x + 8\) while ensuring that each of the provided steps is used only once.

### Step 1:
Use the **Distributive Property**.

\[
2(x + 8) = 2x + 16
\]

Now the equation reads:

\[
2x + 16 = 2x + 8
\]

### Step 2:
Use the **Subtraction of Equality** property to isolate the variable.

Subtract \(2x\) from both sides:

\[
2x + 16 - 2x = 2x + 8 - 2x
\]

This simplifies to:

\[
16 = 8
\]

Since \(16 = 8\) is a false statement, this confirms that there is no solution.

### Conclusion:
For this equation, there are:

**Properties and Reasons Used**:
1. **Distributive Property**
2. **Subtraction of Equality**

Thus, the final determination is:
- There is **no solution** since we reached a contradiction with \(16 = 8\).