Final answer:
Upon evaluating each equation, it is determined that options B (-5x + 12 = 5x - 5) and C (-5x + 12 = -12x - 12) are the ones that have exactly one solution, while options A and D have no solutions.
Step-by-step explanation:
To determine which of the given equations have exactly one solution, we can analyze each equation individually:
- A. -5x + 12 = 5x + 12: Both sides of the equation are equal except for the opposite signs of the x terms. This means that, no matter what value x takes, the x terms will cancel each other out, making the equation incorrect. Hence, this equation has no solutions.
- B. -5x + 12 = 5x - 5: The x terms here can be moved to one side and the constants to the other. Adding 5x to both sides gives -5x + 5x + 12 = 5x + 5x - 5, simplifying to 12 = 10x - 5. Thus, this equation can be solved for x, indicating it has exactly one solution.
- C. -5x + 12 = -12x - 12: Again, we move all x terms to one side and the constants to the other to get -5x + 12x = -12 - 12, simplifying to 7x = -24. Solving for x, we get x = -24/7, indicating that this equation also has exactly one solution.
- D. -5x + 12 = -5x - 12: In this equation, the x terms are identical on both sides, meaning they will cancel out, leaving us with 12 = -12, which is never true. Hence, this equation has no solutions.
From this analysis, we see that options B and C have exactly one solution.