Find the value of k such that the system is inconsistent.

Question

2x+2y=1

9x+ky=-1

Answer 1

Consider the matrix form of this system with:

2 2

9 k

as a matrix of coefficients. The system will have a unique solution when the determinant of this matrix is non-zero so let us find when the determinant IS equal to zero.

(2)(k) - (2)(9) = 0

k=9

Now consider the system again with k=9.

2x + 2y = 1

9x + 9y = -1

Notice that multiplying the first equation by 9/2 we get the following,

9x + 9y = 9/2

9x + 9y = -1

Clearly we cannot have 9x + 9y equal to both 9/2 and -1 hence this will be inconsistent.

Another way to look at this is to think of a value of k such that the slopes of the straight line equations are the same.