Question

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?

Answer 1

Interesting....
the identity result ... 1=1 is a infinite solutions
the contadiction result... 0=1 is a no solution
the exact answer... x=1 is a unique solution


how do we mess with this to get each one?

Answer 2

if k=1
x+y=1
x+y=1
invert one of them to -x-y=-1
and add them
x+y=1
-x-y=-1
0=0 is the identity... so k=1 is an infinite solutions condition.

Answer 3

what if k=-1?

Answer 4

Then there's no solution?

Answer 5

(for k=-1)

Answer 6

correct... 0=2.. no solution

Answer 7

Interesting. Hopefully I can figure out the harder problems from here. Thanks for the help!

Answer 8

no problem... it looks like any other non zero number will get x=-y, which is one unique.... so I guess we are done.