Question 16

More than two

The pair (x, y) = (k, -k) satisfies the equation 2x + 2y = 0 for infinitely many values of k. By substituting x and y into the equation, we can see that it simplifies to an identity that holds true for any value of k.

A) None

This option is incorrect because there are indeed values of k that make the equation true. Since the equation can be satisfied by substituting various values for k, stating that there are "none" is not valid.

B) One

This choice is also incorrect as it suggests that only one specific value of k would satisfy the equation. However, the derived identity allows for multiple values of k, not just one.

C) Two

While this option suggests that exactly two values of k could be solutions, it underestimates the actual number of solutions. The equation allows for any real number k, meaning there are infinitely many solutions, not just two.

D) More than two

This is the correct answer because the equation 2x + 2y = 0 simplifies to 0 = 0 when substituting (k, -k), indicating that it holds true for any real number k. Therefore, there are infinitely many solutions, which is indeed more than two.

Conclusion

The correct answer is "More than two" as the equation is satisfied for all values of k, indicating an infinite number of solutions. Options A, B, and C fail to recognize the nature of the equation and the infinite solutions it provides, thus making them incorrect.