Question 16528
Perhaps a little common sense is required for this problem.

If a line has the same x- and y-intercept, then the line has a negative slope and the slope must be -1.

How can we say this?  If the line had a positive slope the x-intercept would have the opposite sign of the y-intercept, unless, of course, it crossed at the origin.

Now if the x- and y-intercepts are the same value (both positive or both negative), then the line must have a negative slope and, since slope is defined as rise over run, the rise and the run are equal (in absolute value), ergo, the slope is -1.
But how does this find the value of K?

Well, let;s put your equation into the slope-intercept form.

Kx - 2y = 6  Add 2y to both sides.
Kx = 2y + 6  Subtract 6 from both sides.
Kx - 6 = 2y Divide both sides by 2.
y = (K/2)x - 6 The slope, m = K/2 = -1, So K = -2

The given line is:
 y = (-1)x - 6
The x-intercept is: 0 = -x - 6, so x = -6
The y-intercept is: y = 0 - 6,  so y = -6