Q:

For what value of $K$ are the points $A(1, -1)$, $B(K/2, K/3)$, and $C(4, 1)$ collinear?

For what value of $K$ are the points $A(1, -1)$, $B(K/2, K/3)$, and $C(4, 1)$ collinear?

The equation of line $AC$ is $2x-3y=5$. But the point $B$ doesn't satisfy the equation.

A:

"The equation of line AC is 2x-3y=5. But the point B doesnt satisfy the equation."

WHy not?
If that is the equation of the line and if we are told that $(\frac K2, \frac K3)$ is a point of the line, then it must be true that $2*\frac K2 - 3*\frac K3 = 5$.  For what value of $K$ is that true?
1) Is that the equation of the line?
The slope of the equation of the line is $m = \frac {C_y - A_y}{C_x - A_x} = \frac {1-(-1)}{4-1} = \frac 23$  so the equation is $(y-A_y) = m*(x-A_x)$ so $y-(-1) = \frac 23(x-1)$ or $y = \frac 23 x -\frac 53$ or as you put it $2x - 3y = 5$.
2) What value of $K$ satisfies $2*\frac K2 - 3*\frac K3 = 5$ ?
$K - K = 5$
$0 = 5$ 
There is no value of $K$ that satisfies.
So the answer to "For what value of K are the points A(1, -1), B(K/2, K/3), and C(4, 1) collinear?" is...
"None".
Which is a perfectly acceptable and valid answer.  Just because a textbook asks a question doesn't mean that there is an valid answer.  But in this case "none" is a valid answer.  Just because a textbook asks "when does this occur" doesn't mean the answer isn't "never".

