Question

Match the description of the one variable equation with the number of solutions it will have.
x - 7 = 7 - x ____________
3(x + 5) = 3x + 5 ____________
10 - x = 25 ____________
2(x + 4) = 2x + 5 + 3 _____________
Options: One Solution, Two Solutions, No Solution, Infinite Solutions

Answer 1

are you sure this is correct

Answer 2

Let's analyze each equation to determine the number of solutions:

1. **Equation:** \(x - 7 = 7 - x\)
- Rearranging gives: \(x + x = 7 + 7\) → \(2x = 14\) → \(x = 7\)
- **Number of solutions:** One Solution

2. **Equation:** \(3(x + 5) = 3x + 5\)
- Expanding gives: \(3x + 15 = 3x + 5\)
- Rearranging gives: \(15 = 5\), which is false.
- **Number of solutions:** No Solution

3. **Equation:** \(10 - x = 25\)
- Rearranging gives: \(-x = 25 - 10\) → \(-x = 15\) → \(x = -15\)
- **Number of solutions:** One Solution

4. **Equation:** \(2(x + 4) = 2x + 5 + 3\)
- Expanding gives: \(2x + 8 = 2x + 8\)
- This is always true for any \(x\).
- **Number of solutions:** Infinite Solutions

Now, matching the equations with their numbers of solutions:

1. \(x - 7 = 7 - x\) → **One Solution**
2. \(3(x + 5) = 3x + 5\) → **No Solution**
3. \(10 - x = 25\) → **One Solution**
4. \(2(x + 4) = 2x + 5 + 3\) → **Infinite Solutions**

So, summarizing:

- \(x - 7 = 7 - x\) → One Solution
- \(3(x + 5) = 3x + 5\) → No Solution
- \(10 - x = 25\) → One Solution
- \(2(x + 4) = 2x + 5 + 3\) → Infinite Solutions

Answer 3

Let me double-check the equations and my previous responses for accuracy:

1. **Equation:** \(x - 7 = 7 - x\)
- Rearranging gives:
\[
x + x = 7 + 7 \implies 2x = 14 \implies x = 7
\]
- **Number of solutions:** **One Solution**

2. **Equation:** \(3(x + 5) = 3x + 5\)
- Expanding gives:
\[
3x + 15 = 3x + 5
\]
- Rearranging gives:
\[
15 = 5
\]
- This is always false.
- **Number of solutions:** **No Solution**

3. **Equation:** \(10 - x = 25\)
- Rearranging gives:
\[
-x = 25 - 10 \implies -x = 15 \implies x = -15
\]
- **Number of solutions:** **One Solution**

4. **Equation:** \(2(x + 4) = 2x + 5 + 3\)
- Expanding gives:
\[
2x + 8 = 2x + 8
\]
- This is always true for any \(x\).
- **Number of solutions:** **Infinite Solutions**

### Summary:

- **First equation:** \(x - 7 = 7 - x\) → **One Solution**
- **Second equation:** \(3(x + 5) = 3x + 5\) → **No Solution**
- **Third equation:** \(10 - x = 25\) → **One Solution**
- **Fourth equation:** \(2(x + 4) = 2x + 5 + 3\) → **Infinite Solutions**

Based on the analysis above, the initial conclusion is correct. Thank you for prompting a re-evaluation. If you have further questions or need more explanations, feel free to ask!