https://file.aiquickdraw.com/mathbot/file/1727213861411qb4xx56a.jpg

Let's go through each equation one by one to determine which has no solution.

1. **\( 3(2x + 1) - 8x = -2(x + 1) \):**
   \[
   6x + 3 - 8x = -2x - 2
   \]
   Simplifying both sides:
   \[
   (6x - 8x) + 3 = -2x - 2 \Rightarrow -2x + 3 = -2x - 2
   \]
   Adding \(2x\) to both sides:
   \[
   3 = -2
   \]
   This is a contradiction, meaning **this equation has no solution**.

2. **\( 6(2x - 1) = 4(3x - 1) - 2 \):**
   \[
   12x - 6 = 12x - 4 - 2
   \]
   Simplifying both sides:
   \[
   12x - 6 = 12x - 6
   \]
   This is a true statement for all values of \(x\), so this equation has **infinitely many solutions**.

3. **\( 2(2x - 5) = 2(2x - 5) \):**
   Expanding both sides:
   \[
   4x - 10 = 4x - 10
   \]
   This is also a true statement for all \(x\), meaning it has **infinitely many solutions**.

4. **\( 10 - 4(x + 9) = -2(2x + 13) \):**
   Expanding both sides:
   \[
   10 - 4x - 36 = -4x - 26
   \]
   Simplifying:
   \[
   -4x - 26 = -4x - 26
   \]
   This is also true for all \(x\), so this equation has **infinitely many solutions**.

### Conclusion:
The equation with no solution is the first one:
\[
3(2x + 1) - 8x = -2(x + 1)
\]

Would you like further details or have any questions? Here are some related questions:

1. What are the different types of solutions for linear equations?
2. Can a linear equation have more than one solution?
3. How do we recognize a contradiction in a system of equations?
4. What is the difference between inconsistent and consistent systems?
5. How can graphing help visualize solutions to equations?

**Tip:** When solving equations, always look for contradictions like \(3 = -2\) to identify equations with no solution.

Learn how to identify which equation has no solution using fundamental algebraic techniques such as distribution and contradiction in equations. This problem involves linear equations and algebraic manipulation, suitable for Grades 8-10.

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