Which of the following equations have exactly one solution? Choose all answers that apply: (A) \( -5 x+12=-12 x-12 \) (B) \( -5 x+12=-5 x-12 \) (c) \( -5 x+12=5 x+12 \) (D) \( -5 x+12=5 x-5 \)

To determine if an equation has exactly one solution, we need to simplify each equation and check whether it reduces to a statement like "x equals a number" (one solution), a contradiction (no solution), or an identity (infinitely many solutions). Let’s analyze each equation: Equation (A): −5x + 12 = −12x − 12 1. Add 12x to both sides:   7x + 12 = −12 2. Subtract 12:   7x = −24 3. Divide by 7:   x = −24/7 Since this yields a specific value for x, there is exactly one solution. Equation (B): −5x + 12 = −5x − 12 1. Add 5x to both sides:   12 = −12 This is a contradiction (12 does not equal −12), so there is no solution. Equation (C): −5x + 12 = 5x + 12 1. Add 5x to both sides:   12 = 10x + 12 2. Subtract 12:   0 = 10x 3. Divide by 10:   x = 0 This yields exactly one solution. Equation (D): −5x + 12 = 5x − 5 1. Add 5x to both sides:   12 = 10x − 5 2. Add 5:   17 = 10x 3. Divide by 10:   x = 17/10 This also gives exactly one solution. Thus, the equations that have exactly one solution are (A), (C), and (D).