X+ ky = 1
kx + y = 1

how do you find the value(s) of k that give this system (a) no solution, (b) a unique solution, and (c) infinitely many solutions?

X + k y = 1
k x + y = 1  / * ( - k )

      x + k y = 1
- k² x - k y = - k

x - k² x = 1 - k
x ( 1 - k² ) = 1 - k
x = ( 1 - k ) / ( 1 - k² ) = ( 1 - k ) / ( 1 - k ) ( 1 + k )
y = 1 - k( 1 - k )/( 1 - k² )
y = ( 1 - k ) / ( 1 - k² ) = ( 1 - k ) / ( 1 - k ) ( 1 + k )
a ) For  k = - 1 this system has no solution.
b ) For k ≠ - 1 and k ≠ 1, the system has unique solution:
( x , y ) = ( 1/ (1 + k) ,  1/( 1 + k ) ).
c ) For k = 1, there are infinitely many solutions.

There is no association.