Solve the system: x + y = 1 and x − y = 0. How many solutions are there?

Solve the system: x + y = 1 and x − y = 0. How many solutions are there?

• A. No solution
• B. The lines are parallel
• C. One solution
• D. Infinitely many solutions

Answer: (C)

Two lines in the plane either cross at one point, run parallel, or are the same line. Here, adding the two equations together eliminates y, giving 2x = 1, so x = 1/2. Substituting back into x + y = 1 yields y = 1 − 1/2 = 1/2. So there is exactly one solution, the point (1/2, 1/2). This happens because the lines have different slopes (one is y = −x + 1 and the other is y = x), so they intersect at a single point rather than being parallel or identical.

Two lines in the plane either cross at one point, run parallel, or are the same line. Here, adding the two equations together eliminates y, giving 2x = 1, so x = 1/2. Substituting back into x + y = 1 yields y = 1 − 1/2 = 1/2. So there is exactly one solution, the point (1/2, 1/2). This happens because the lines have different slopes (one is y = −x + 1 and the other is y = x), so they intersect at a single point rather than being parallel or identical.