Let's analyze each equation to determine which one has exactly one solution.

A) $3x + 9 = 3(x + 10)$

Distribute the 3 on the right side: $3x + 9 = 3x + 30$ Subtract $3x$ from both sides: $9 = 30$ This is a contradiction, meaning there is no solution.

B) $2(7x - 3) = 15x + 1$

Distribute the 2 on the left side: $14x - 6 = 15x + 1$ Subtract $14x$ from both sides: $-6 = x + 1$ Subtract 1 from both sides: $x = -7$ This equation has one solution: $x = -7$.

C) $\frac{1}{2}(6x - 2) = 3(x - \frac{1}{3})$

Distribute $\frac{1}{2}$ and 3: $3x - 1 = 3x - 1$ Since both sides are the same, this equation has infinite solutions.

D) $2(4x + 3) = 5x + 3(x - 2)$

Distribute the 2 and the 3: $8x + 6 = 5x + 3x - 6$ Simplify: $8x + 6 = 8x - 6$ Subtract $8x$ from both sides: $6 = -6$ This is a contradiction, meaning there is no solution.

Conclusion:

The correct answer is B, as it is the only equation with exactly one solution.

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. What is the difference between an equation with one solution and no solution?
2. How can you recognize when an equation has infinite solutions?
3. How does distributing terms affect the solvability of equations?
4. What methods can be used to solve linear equations with one variable?
5. How can you verify if an equation has no solutions?

Tip: Always simplify both sides of an equation before deciding on the number of solutions!