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The problem asks to determine which linear equation has no solution. Let's analyze each equation one by one:

### Equation 1: 
\[
\frac{2}{3}(9x + 6) = 6x + 4
\]
Distribute the \(\frac{2}{3}\):
\[
\frac{2}{3} \times 9x = 6x, \quad \frac{2}{3} \times 6 = 4
\]
So the equation becomes:
\[
6x + 4 = 6x + 4
\]
This is always true (an identity), so it has infinitely many solutions, not no solution.

### Equation 2:
\[
5x + 12 = 5x - 7
\]
Subtract \(5x\) from both sides:
\[
12 = -7
\]
This is a contradiction (false statement), so this equation **has no solution**.

### Equation 3:
\[
4x + 7 = 3x + 7
\]
Subtract \(3x\) from both sides:
\[
x + 7 = 7
\]
Subtract \(7\) from both sides:
\[
x = 0
\]
This equation has a solution: \(x = 0\).

### Equation 4:
\[
-3(2x - 5) = 15 - 6x
\]
Distribute the \(-3\):
\[
-6x + 15 = 15 - 6x
\]
Add \(6x\) to both sides:
\[
15 = 15
\]
This is an identity, so it has infinitely many solutions.

### Conclusion:
The equation that has **no solution** is **Equation 2: \(5x + 12 = 5x - 7\)**.

Would you like further details or explanations?

### Related Questions:
1. What is meant by a linear equation having no solution?
2. Can an equation have infinitely many solutions, and why?
3. How do you determine when a linear equation has a unique solution?
4. What other forms of contradictions can occur in linear equations?
5. How would graphing help in visualizing equations with no solution?

### Tip:
When both sides of an equation simplify to a false statement (like \(12 = -7\)), the equation has no solution.

Discover which linear equation has no solution by exploring algebraic concepts such as identities and contradictions. This math problem is suitable for Grades 7-9 and involves solving linear equations step-by-step.

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