https://file.aiquickdraw.com/mathbot/file/1726672022835r0glum3u.jpg

Let's go through each equation from the image and solve it step-by-step to match it with the correct solution.

1. **Equation:** \( 4 + x = -8x - 5 \)
   - Combine like terms: 
     \[
     x + 8x = -5 - 4
     \]
     \[
     9x = -9
     \]
     \[
     x = -1
     \]
   - **Solution:** \( x = -1 \)

2. **Equation:** \( 2(5x + 2) = 10x - 4 \)
   - Expand both sides: 
     \[
     10x + 4 = 10x - 4
     \]
   - Subtract \( 10x \) from both sides:
     \[
     4 = -4
     \]
   - This is a contradiction, so there is **no solution**.

3. **Equation:** \( 7 + 2x = 2x - 7 \)
   - Subtract \( 2x \) from both sides:
     \[
     7 = -7
     \]
   - This is also a contradiction, so there is **no solution**.

4. **Equation:** \( -3x + 3 = 3(1 - x) \)
   - Expand the right-hand side:
     \[
     -3x + 3 = 3 - 3x
     \]
   - Add \( 3x \) to both sides:
     \[
     3 = 3
     \]
   - This is always true, meaning the solution is \( x = \) **all real numbers**.

### Final Matching:

1. \( 4 + x = -8x - 5 \) → \( x = -1 \)
2. \( 2(5x + 2) = 10x - 4 \) → **No solution**
3. \( 7 + 2x = 2x - 7 \) → **No solution**
4. \( -3x + 3 = 3(1 - x) \) → \( x = \) **all real numbers**

Would you like further explanations or more details on any of the steps?

Here are 5 related questions:
1. How do you determine when an equation has no solution?
2. Can an equation have more than one solution?
3. What does it mean for an equation to have "all real numbers" as a solution?
4. How would you solve more complex multi-variable equations?
5. What are the different types of contradictions in equations?

**Tip:** Always check your final solution by substituting it back into the original equation to verify correctness.

Match each equation on the left to its solution on the right. Some answer choices on the right will be used more than once.

Learn how to solve linear equations and inequalities by matching equations to their solutions. This problem involves key concepts like no solution, infinite solutions, and properties of equality. Suitable for Grades 7-10 students.

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