Question 19

There is only 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 equal zero, as a sum of squares is zero only when each square is zero. This leads to the conclusion that there is precisely one solution for the pair (x, y).

A) None

This option is incorrect because it implies that there are no solutions to the equation. However, since the equation can be satisfied by specific values of x and y, this option does not hold true.

B) One only

This option is correct as it accurately reflects that there is exactly one solution to the equation. Setting each squared term to zero gives us the unique solution: x = 7 and y = -3.

C) Two only

This option is incorrect because it suggests that there are two distinct pairs of values (x, y) that satisfy the equation. The nature of the equation, being a sum of squares equal to zero, indicates that only one specific pair can satisfy it.

D) Three only

This option is also incorrect as it implies there are three pairs of values (x, y) that satisfy the equation. Like the previous options, this misinterpretation does not consider that a sum of squares can only be zero for a unique set of values.

E) More than three

This option is incorrect since it suggests the existence of more than three pairs of values (x, y) that satisfy the equation. The equation's structure only allows for one unique solution.

Conclusion

The only correct answer is that there is one pair of numbers (x, y) that satisfies the equation (x − 7)² + (y + 3)² = 0, specifically (7, -3). All other options fail because they either overestimate the number of solutions or incorrectly claim there are no solutions at all.