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.