Question

Consider the given system of linear equations for variables \(x\) and \(y\), where \(k\) is a real-valued constant. Which of the following option(s) is/are CORRECT? \[ x + ky = 1 \] \[ kx + y = -1 \]

• A. There is exactly one value of \(k\) for which the above system of equations has no solution.
• B. There exist an infinite number of values of \(k\) for which the system of equations has no solution.
• C. There exists exactly one value of \(k\) for which the system of equations has exactly one solution.
• D. There exists exactly one value of \(k\) for which the system of equations has an infinite number of solutions.

Hint:

When solving systems of equations, check the determinant of the coefficient matrix to determine if the system has a unique solution, no solution, or infinitely many solutions.

Answer 1

The Correct Option is A, D

For this system, the determinant of the coefficient matrix is \( 1 - k^2 \).
For the system to have no solution, we set the determinant equal to zero: \[ 1 - k^2 = 0 \quad \Rightarrow \quad k = \pm 1. \] Hence, for \( k = 1 \), the system has no solution. For \( k \neq 1 \), the system has exactly one solution. For \( k = -1 \), the system has infinitely many solutions. Therefore, the correct answer is \( \boxed{A} \) \& \( \boxed{D} \).