y-k=5(x-h)/3 represents all lines with a slope of 5/3. When x=h, y=k.
So, if h and k are both zero, y=5x/3 passing through the origin. If h=1 and k=1, y-1=5(x-1)/3, y=5x/3-5/3+1=5x/3-2/3 or (5x-2)/3. The y-intercept is -2/3 and the x-intercept is 2/5. So the general equation for all lines with the slope 5/3 is:
3y-3k=5x-5h, that is, 5x-3y+3k-5h=0.
For slope -2/7, we have y-k=-2(x-h)/7, 7y-7k=-2x+2h, 2x+7y-7k-2h=0.
For undefined slope (infinite): vertical line: x=h.
Slope 4: y=4(x-h)+k, y=4x-4h+k or 4x-y+k-4h=0.
Slope -8: y=-8(x-h)+k, y=-8x+8h+k, 8x+y-k-8h=0.
Slope 0: horizontal line: y=k.
Note that h and k can be any numbers in all cases, so (h,k) represents the points that the lines pass through.