To determine which of Aanya’s equations has exactly one solution, we need to analyze each equation for the possibility of having one, no, or infinitely many solutions.
Equation A:
\[ 6x - 8 = 4(x - 2) + 2x \] Simplifying the right side: \[ 6x - 8 = 4x - 8 + 2x \] Combining like terms on the right: \[ 6x - 8 = 6x - 8 \] This simplifies to a true statement, which indicates that there are infinitely many solutions.
Equation B:
\[ 3(x - 1) + 2x = 3(x - 1) + 2 \] Simplifying both sides: \[ 3x - 3 + 2x = 3x - 3 + 2 \] Combining like terms: \[ 5x - 3 = 3x - 3 + 2 \] Which simplifies to: \[ 5x - 3 = 3x - 1 \] Subtracting \(3x\) from both sides: \[ 2x - 3 = -1 \] Adding 3 to both sides: \[ 2x = 2 \] Dividing by 2: \[ x = 1 \] This equation has exactly one solution.
Equation C:
\[ 7x + 2 - x = 6(x + 2) \] Simplifying the left side: \[ 6x + 2 = 6x + 12 \] Subtracting \(6x\) from both sides: \[ 2 = 12 \] This is a false statement, indicating that there is no solution.
Equation D:
\[ 4(x + 3) + x = 5(x + 1) + 7 \] Expanding both sides: \[ 4x + 12 + x = 5x + 5 + 7 \] Combining like terms: \[ 5x + 12 = 5x + 12 \] This also simplifies to a true statement, indicating there are infinitely many solutions.
Conclusion:
The only equation that has exactly one solution is Equation B:
\[ 3(x - 1) + 2x = 3(x - 1) + 2 \]
1 answer