The standard form of a linear equation is:
#color(red)(A)x + color(blue)(B)y = color(green)(C)#
where, if at all possible, #color(red)(A)#, #color(blue)(B)#, and #color(green)(C)#are integers, and A is non-negative, and, A, B, and C have no common factors other than 1
The first step is to multiple each side of the equation by #color(red)(3)# to obtain all integers:
#color(red)(3) xx y = color(red)(3) xx (x/3 - 6)#
#3y = (color(red)(3) xx x/3) - (color(red)(3) xx 6)#
#3y = (cancel(color(red)(3)) xx x/color(red)(cancel(color(black)(3)))) - 18#
#3y = x - 18#
Next step is to move #x# to the left side of the equation by subtracting #color(red)(x)# from each side of the equation:
#-x + 3y = 0 - 18#
#-x + 3y = -18#
Now we can multiply each side of the equation by #color(red)(-1)# to make the coefficient of #x# positive. The coefficient is #-1# currently.
#color(red)(-1) xx (-x + 3y) = color(red)(-1) xx -18#
#x - 3y = 18#
or
#color(red)(1)x - color(blue)(3)y = color(green)(18)#
