Question

for what values of k does the linear system below have:
a) infinite solutions?
b) one solution?
c) no solution?

2/3x+y=16
kx+3y=48

Answer 1

To have an infinite number of solutions, both equations are really the same equation

if we multiply the first by 3,
2x + 3y = 48
comparing with the 2nd:
kx + 3y = 48
there would be an infinite number of solutions if k = 2

One solution, k ≠ 2

to have no solution, the two lines would have to be parallel, that is, they must have the same slope
so (2/3) / 1 = k/3
k = 2, but we found that if k = 2, they are one and the same line, with an infinite number of solutions.
Thus no value of k exists for which there would be no solution