From the equations where z = x/y, if you obtain two conflicting values for z, the system has how many solutions?

• A. One solution
• B. Infinitely many solutions
• C. Two solutions
• D. Zero solutions

Answer: (D)

When a system forces a variable to equal two different numbers, there’s no way to satisfy both conditions at once. Here, z is defined as x over y, so z is determined by x and y (with y not zero). If the system requires z to take two conflicting values, no pair (x, y) can make z equal both values simultaneously. That makes the whole system inconsistent, so there are no solutions. If any part of the setup tried to force y to be zero, z would be undefined, which also rules out solutions. So the situation yields zero solutions.