Question 7

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 satisfied when both squared terms are equal to zero. This leads to a single solution for the pair of numbers.

A) None

This option is incorrect because the equation does have a solution. The sum of two squares can equal zero only when both squares are individually zero, indicating that at least one pair of numbers exists.

B) One only

This option is correct. The equation \((x - 7)^2 + (y + 3)^2 = 0\) implies that both \((x - 7)^2\) and \((y + 3)^2\) must equal zero simultaneously, leading to the solution \(x = 7\) and \(y = -3\), which is precisely one unique pair.

C) Two only

This option is incorrect. The equation cannot yield two distinct pairs as it requires both components of the equation to be zero, thus providing only one unique solution rather than multiple.

D) Three only

This option is incorrect. Similar to option C, the requirement that both parts of the equation equal zero restricts the solution to just one pair, making the possibility of three pairs impossible.

E) More than three

This option is incorrect. The nature of the equation, being a sum of squares equal to zero, permits only one valid solution, thereby negating the existence of more than three pairs.

Conclusion

The only valid solution from the equation \((x - 7)^2 + (y + 3)^2 = 0\) is when both squared terms are zero, leading to a unique pair of numbers \( (7, -3) \). All other options fail to recognize the fundamental property of squares that dictates the requirement for both terms to equal zero simultaneously.