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)
