Which statement accurately describes a system with a single solution in the given examples?

Which statement accurately describes a system with a single solution in the given examples?

• A. No solution
• B. Infinite solutions
• C. One solution: x = 5 or y = 5; no solution: 5 = 4; infinite solutions: 5 = 5
• D. All three are true for different systems

Answer: (C)

When solving a system of equations, you check how many solutions the pair of lines represents: one intersection point means a single solution, parallel but distinct lines mean no solution, and the same line means infinitely many solutions. In the examples, reaching a concrete value for a variable—such as x = 5 or y = 5—shows you’ve pinpointed the exact coordinates of the intersection, and then the other variable can be found to complete the unique pair. A statement like 5 = 4 would contradict the equations, signaling there is no possible solution. A statement like 5 = 5 indicates the equations describe the same line, so there are infinitely many solutions. Therefore, describing a system with exactly one solution as having a definite value for a variable best captures the idea of a unique intersection.

When solving a system of equations, you check how many solutions the pair of lines represents: one intersection point means a single solution, parallel but distinct lines mean no solution, and the same line means infinitely many solutions. In the examples, reaching a concrete value for a variable—such as x = 5 or y = 5—shows you’ve pinpointed the exact coordinates of the intersection, and then the other variable can be found to complete the unique pair. A statement like 5 = 4 would contradict the equations, signaling there is no possible solution. A statement like 5 = 5 indicates the equations describe the same line, so there are infinitely many solutions. Therefore, describing a system with exactly one solution as having a definite value for a variable best captures the idea of a unique intersection.