Question

The number of values of k, for which the system
of equations
(k + 1)x + 8y = 4k
kx + (k + 3)y = 3k
−
1
has no solution, is
(1) 1
(2) 2
(3) 3
(4) infinite

Answer 1

i think 2

Answer 2

For them to have no solution, the lines described by the equations must be parallel, i.e. $$k+1=kt\\8=(k+3)t\\4k=3kt$$How many \(k\) satisfy the above constraints for arbitrary \(t\)?

Answer 3

$$4k=3(k+1)\\4k=3k+3\\k=3\\t=\frac43$$... this is the only solution I can find.

Answer 4

It's an overdetermined system anyways.