For how many pairs of numbers x and y does (x − 7)² + (y + 3)² = 0?

Rationale

There is one pair of numbers x and y that satisfies the equation.

The equation \((x - 7)^2 + (y + 3)^2 = 0\) can only be true if both squared terms are equal to zero. This condition leads to a unique solution for \(x\) and \(y\), specifically \(x = 7\) and \(y = -3\).

A (None) YOUR ANSWER

There is at least one solution to the equation, as both squared terms can be zero simultaneously. Thus, this option is incorrect because it dismisses the existence of any valid pairs (x, y).

B (One only) CORRECT

This is the correct choice since the only solution occurs when both \((x - 7)^2 = 0\) and \((y + 3)^2 = 0\), resulting in the unique pair \((x, y) = (7, -3)\). Hence, there is precisely one pair of numbers that satisfies the equation.

C (Two only) YOUR ANSWER

Two pairs would imply there are two distinct solutions for the equation, which is not the case here. Since both squared components must equal zero simultaneously for the equation to hold true, only one solution exists.

D (Three only) YOUR ANSWER

Three distinct pairs would suggest additional solutions, but the squared terms restrict the equation to a single solution. Thus, this option incorrectly implies multiple pairs exist.

E (More than three) YOUR ANSWER

This choice also incorrectly suggests there are more than one solution pairs. Given the nature of the equation, where both terms must equal zero, there can only be one solution pair.

Conclusion

The equation \((x - 7)^2 + (y + 3)^2 = 0\) leads to a unique solution, confirming that there is only one pair of numbers, \( (7, -3) \), that satisfies it. All other options incorrectly suggest the existence of multiple solutions, which contradicts the foundational requirement that both squared terms must equal zero simultaneously.

For how many pairs of numbers x and y does (x − 7)² + (y + 3)² = 0?

There is one pair of numbers x and y that satisfies the equation.

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