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, multiply each side of the equation by #color(red)(2)# to ensure all coefficients are integers:
#color(red)(2)(y - 4) = color(red)(2) xx 2.5(x + 3)#
#2y - 8 = 5(x + 3)#
Next, expand the terms on the right side of the equation by multiplying each term within the parenthesis by the term outside the parenthesis:
#2y - 8 = color(red)(5)(x + 3)#
#2y - 8 = (color(red)(5)xx x) + (color(red)(5)xx 3)#
#2y - 8 = 5x + 15#
Then, add #color(red)(8)# and subtract #color(blue)(5x)# from each side of the equation to place the #x# and #y# terms on the left side of the equation and constant on the right side of the equation while keeping the equation balanced:
#-color(blue)(5x) + 2y - 8 + color(red)(8) = -color(blue)(5x) + 5x + 15 + color(red)(8)#
#-5x + 2y - 0 = 0 + 23#
#-5x + 2y = 23#
Now, multiply each side of the equation by #color(red)(-1)# to ensure the #x# coefficient is positive while keeping the equation balanced:
#color(red)(-1)(-5x + 2y) = color(red)(-1) xx 23#
#(color(red)(-1) xx -5x) + (color(red)(-1) xx 2y) = -23#
#color(red)(5)x - color(blue)(2)y = color(green)(-23)#
