Consider the given system of linear equations for variables \(x\) and \(y\),…

Compute the determinant of the coefficient matrix to classify the system.

Determinant = 1 - k^2. If this determinant is nonzero (k ≠ ±1) the system has a unique solution; if it is zero (k = ±1) investigate consistency.

• Case k = 1: The equations become x + y = 1 and x + y = -1. These are inconsistent (they assert two different values for x + y), so there is no solution.

• Case k = -1: The equations become x - y = 1 and -x + y = -1. The second equation is just −1 times the first, so the two equations are dependent and the system has infinitely many solutions (one free parameter). For example, set y = t, then x = 1 + t.

• Case k ≠ ±1: The determinant is nonzero, so the system has exactly one solution. Using Cramer's rule gives x = 1/(1 - k) and y = -1/(1 - k), which are well defined for k ≠ ±1.

Conclusion: The system has no solution only at k = 1; it has infinitely many solutions only at k = -1; and it has a unique solution for every other real k. Therefore the correct statements are those that assert exactly one value of k gives no solution (k = 1) and exactly one value of k gives infinitely many solutions (k = -1).