To determine which equation has exactly one solution, we analyze each given equation step by step.

A. $3x = x + 2x$

Simplify the right-hand side: $x + 2x = 3x$ So, the equation becomes: $3x = 3x$ This is true for all values of $x$, which means it has infinitely many solutions.

B. $3(x + 1) = 3x + 3$

Distribute $3$ on the left-hand side: $3x + 3 = 3x + 3$ This equation simplifies to: $3x + 3 = 3x + 3$ This is true for all values of $x$, so it also has infinitely many solutions.

C. $3x = x + 2$

Rearrange terms to bring all $x$-terms to one side: $3x - x = 2$ Simplify: $2x = 2$ Solve for $x$: $x = 1$ This equation has exactly one solution.

D. $3(x + 1) = 3x + 2$

Distribute $3$ on the left-hand side: $3x + 3 = 3x + 2$ Simplify: $3x + 3 - 3x = 2$ This reduces to: $3 = 2$ This is a false statement, meaning there are no solutions.

Final Answer:

The equation that has exactly one solution is:

C. $3x = x + 2$

Do you need further explanation or clarification? Let me know!

Here are some related questions to explore further:

1. What does it mean for an equation to have no solutions?
2. How do you determine if an equation has infinitely many solutions?
3. Can you solve equations involving more variables, like $2x + 3y = 5$?
4. What happens if the coefficients on both sides of the equation are equal?
5. How do we check if a solution satisfies an equation?

Tip:

Always simplify both sides of the equation and combine like terms to determine the nature of the solutions.