Question

Find all real constants ( k ) such that the system ( x + 3y = kx ) and ( 3x + y = ky ) when ( (x, y) ) has a solution other than ( (x, y) = (0, 0) ).

A. ( k = 3 )
B. ( k = 2 )
C. ( k = 1 )
D. ( k = 0 )

Answer 1

To find the values of 'k' for which the system of equations has a solution other than (x, y) = (0, 0), we can solve the system of equations and determine when it is consistent (has infinite solutions). The values of 'k' are 3 and -3.

Step-by-step explanation:

To find the values of k for which the system of equations has a solution other than (x, y) = (0, 0), we can solve the system of equations and determine when it is consistent (has infinite solutions).

First, let's rewrite the system of equations in matrix form:

[1 - k, 3; 3, 1 - k] [x; y] = [0; 0]

The system has a solution other than (x, y) = (0, 0) if and only if the determinant of the coefficient matrix is zero:

(1 - k)(1 - k) - 9 = 0

Simplifying the equation, we get:

(k - 3)(k + 3) = 0

So, the values of k for which the system has a solution other than (x, y) = (0, 0) are: k = 3 and k = -3.