Find conditions on k that will make the following system of equations have a unique solution. To enter your answer, first select whether k should be equal or not equal to specific values, then enter a value or a list of values separated by commas.

Question

Then give a formula in terms of k for the solution to the system, when it exists. Be sure to include parentheses where necessary, e.g. to distinguish 1/(2k) from 1/2k.

kx+y = 5
2x+2ky = 3

The system has a unique solution when k = ???
The unique solution is xy = 0
0

Answer 1

The slopes must be different to have a unique solution. So, we need

-k ≠ -1/k
k^2 ≠ 1

To find that solution, multiply the 1st equation by 2k and you have

2k^2 x + 2ky = 10k
2x + 2ky = 3

Now subtract to eliminate y, and you have

(2k^2-2)x = 10k-3
x = (10k-3)/(2k^2-2)

Substituting that in, we have

k(10k-3)/(2k^2-2) + y = 5
y = (3k-10)/(2k^2-2)