Linear Algebra

Step 1:
The given system of equations is:
We need to find the values of and for which this system has a unique solution.
Step 2:
The condition for a unique solution in a system of linear equations is that the determinant of the coefficient matrix should be non-zero. The coefficient matrix is: The determinant of this matrix is:
Step 3:
Calculate each of the 2x2 determinants:
Step 4:
Substitute these values back into the determinant expression:
Step 5:
For the system to have a unique solution, the determinant must not be zero:
Step 6:
Substitute into any equation, say equation (3), and solve for : From the other equations, it follows that .
Step 7:
Therefore, the values for and for which the system has a unique solution are and .