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
First, to convert the equation in the problem to standard form we can multiply each side of the equation by #color(red)(3)# to eliminate the fractions as required by the formula while keeping the equation balanced:
#color(red)(3) * y = color(red)(3)(2/3x - 7)#
#3y = (color(red)(3) * 2/3x) - (color(red)(3) * 7)#
#3y = (cancel(color(red)(3)) * 2/color(red)(cancel(color(black)(3)))x) - 21#
#3y = 2x - 21#
Next, we can subtract #color(red)(2x)# from each side of the equation to isolate the #x# and #y# term on the left side of the equation as required by the formula while keeping the equation balanced:
#-color(red)(2x) + 3y = -color(red)(2x) + 2x - 21#
#-2x + 3y = 0 - 21#
#-2x + 3y = -21#
Now, multiply each side of the equation by #color(red)(-1)# to convert the #x# coefficient to a positive integer as required in the formula while keeping the equation balanced:
#color(red)(-1)(-2x + 3y) = color(red)(-1) * -21#
#(color(red)(-1) * -2x) + (color(red)(-1) * 3y) = 21#
#color(red)(2)x - color(blue)(3)y = color(green)(21)#
