1. **State the problem:** We need to determine which of the given linear equations have no solutions. 2. **Recall the rule:** A linear equation has no solution if, after simplifying, it results in a contradiction such as $a = b$ where $a \neq b$ and there is no variable term left. 3. **Analyze each equation:** - A. $4x + 3 = 3x + 1$ Subtract $3x$ from both sides: $4x - 3x + 3 = 1 \Rightarrow x + 3 = 1$ Subtract 3: $x = -2$ Has a solution. - B. $5x + 2 = 5x + 8$ Subtract $5x$: $2 = 8$ This is a contradiction, so no solution. - C. $3(x + 4) = 3x + 12$ Expand left: $3x + 12 = 3x + 12$ Subtract $3x + 12$: $0 = 0$ Infinite solutions, so it has solutions. - D. $2(4x + 2) = 8x + 9$ Expand left: $8x + 4 = 8x + 9$ Subtract $8x$: $4 = 9$ Contradiction, no solution. - E. $4(x + 3) = 10$ Expand left: $4x + 12 = 10$ Subtract 12: $4x = -2$ Divide by 4: $x = -\frac{1}{2}$ Has a solution. 4. **Final answer:** Equations B and D have no solutions.