First, expand the terms in parenthesis on the right side of the equation by multiplying each term within the parenthesis by the term outside the parenthesis:
#4y - 5x = color(red)(3)(4x - 2y + 1)#
#4y - 5x = (color(red)(3) * 4x) - (color(red)(3) * 2y) + (color(red)(3) * 1)#
#4y - 5x = 12x - 6y + 3#
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.
To convert to the Standard Form of a linear equation we need to first subtract #color(red)(12x)# and add #color(blue)(6y)# to each side of the equation to get the #x# and #y# terms on left side of the equation while keeping the equation balanced:
#-color(red)(12x) + color(blue)(6y) + 4y - 5x = -color(red)(12x) + color(blue)(6y) + 12x - 6y + 3#
#-color(red)(12x) - 5x + color(blue)(6y) + 4y = -color(red)(12x) + 12x + color(blue)(6y) - 6y + 3#
#(-color(red)(12) - 5)x + (color(blue)(6) + 4)y = 0 - 0 + 3#
#-17x + 10y = 3#
Now, multiply each side of the equation by #color(red)(-1)# to ensure the coefficient of the #x# is a positive integer:
#color(red)(-1)(-17x + 10y) = color(red)(-1) * 3#
#(color(red)(-1) * -17x) + (color(red)(-1) * 10y) = -3#
#17x + (-10y) = -3#
#color(red)(17)x - color(blue)(10)y = color(green)(-3)#
