Introduction:
In this problem, we are asked to find the value of K for which the system of equation have infinity many x-Ky, 3x 6y = -5.

Step 1: Convert the system of equation to slope-intercept form:
To solve this problem, we first need to convert the given system of equation into slope-intercept form. The slope-intercept form of an equation is y = mx + b, where m is the slope and b is the y-intercept.

Step 2: Solve for y:
We can solve the given system of equation for y by rearranging the terms as follows:

3x - 6y = -5
-6y = -3x - 5
y = (1/2)x + 5/6

Step 3: Find the value of K:
Now, we can rewrite the equation y = (1/2)x + 5/6 as y - (1/2)x = 5/6. This equation is in the form y - mx = b, where m = 1/2 and b = 5/6.

To have infinitely many solutions, the line x - Ky must be parallel to the line y = (1/2)x + 5/6. This means that the slope of x - Ky must be equal to the slope of y = (1/2)x + 5/6.

The slope of x - Ky is -K, so we have:

-K = 1/2

Solving for K, we get:

K = -1/2

Conclusion:
Therefore, the value of K for which the system of equation have infinitely many solutions of x - Ky is -1/2.