The system of equations with z = x/y, described as (x/y) - 12 + 14 = 36 and (x/y) - 6 + 7 = -18 has how many solutions?

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Multiple Choice

The system of equations with z = x/y, described as (x/y) - 12 + 14 = 36 and (x/y) - 6 + 7 = -18 has how many solutions?

Explanation:
The key idea is that the same ratio x/y, which is defined as z, must take on a single value in both equations. If you isolate the ratio in each equation, you get two different results: from the first, x/y = 34; from the second, x/y = -19. Since z = x/y, z would have to be 34 and -19 at the same time, which is impossible. Also, for the ratio to be defined, y cannot be zero, but that doesn’t resolve the contradiction—there’s no value of x/y that satisfies both equations simultaneously. So there are no solutions.

The key idea is that the same ratio x/y, which is defined as z, must take on a single value in both equations. If you isolate the ratio in each equation, you get two different results: from the first, x/y = 34; from the second, x/y = -19. Since z = x/y, z would have to be 34 and -19 at the same time, which is impossible. Also, for the ratio to be defined, y cannot be zero, but that doesn’t resolve the contradiction—there’s no value of x/y that satisfies both equations simultaneously. So there are no solutions.

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