For how many values of k is (x, y) = (k, -k) a solution to the equation 2x +2y = 0?

Rationale

There are more than two values of k for which (x, y) = (k, -k) is a solution to the equation 2x + 2y = 0.

Substituting (x, y) = (k, -k) into the equation yields 2k + 2(-k) = 0, which simplifies to 0 = 0. This identity holds true for any real number k, indicating that there are infinitely many values of k that satisfy the equation.

A (None) YOUR ANSWER

This choice suggests that there are no values of k that work, which is incorrect. The derived equation 2k + 2(-k) = 0 is always true, meaning there are indeed values of k that satisfy the equation.

B (One) YOUR ANSWER

This option implies that there is only a single value of k that makes (x, y) a solution. However, since the equation simplifies to a true statement for any k, this option is false. There is not just one but an infinite number of solutions.

C (Two) YOUR ANSWER

This choice indicates that there are exactly two values of k satisfying the equation. However, since we have established that the equation is satisfied by any real number k, suggesting only two solutions is incorrect; there are infinitely many.

D (More than two) CORRECT

This option correctly states that there are more than two values of k that solve the equation. Since (k, -k) is valid for any real number k, this statement encompasses the infinite nature of the solutions.

Conclusion

The equation 2x + 2y = 0 allows for infinitely many pairs (k, -k) as solutions. Thus, the correct answer indicates that there are more than two values of k that satisfy the equation, confirming the extensive solution set inherent in linear equations of this form.

For how many values of k is (x, y) = (k, -k) a solution to the equation 2x +2y = 0?

There are more than two values of k for which (x, y) = (k, -k) is a solution to the equation 2x + 2y = 0.

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