3x = 2y + 7

I know the answer: x=9 and y=10, but what are the algebraic steps to solve the equation?




If you have one equation with two unknowns then there may be many solutions. In your example you give the solution x = 9 and y = 10 but there are other solutions. For example x = 3 and y = 1, or x = 5 and y = 4. In fact I'll try to convince you that there are infinitely many integer solutions.

In your equation

3x = 2y + 7

the left side is divisible by 3 so the right side must be divisible by 3. If 7 is divided by 3 it leaves a remainder of 1 so, since the right side is divisible by 3, 2y divided by 3 must have a remainder of 2. That means that y divided by 2 has a remainder of 1. In other words, y must be one more than a multiple of 3, that is y can be 1, 4, 7, 10, 13,...

For each such choice of y, both sides of the equation are divisible by 3 and hence if you divide both sides by 3 you get an integer value for x, and this value of x, together with the chosen value of y is a solution to the equation.

For example if y = 52 (1 more than a multiple of 3) then 2y + 7 = 111 and 111/3 = 37 so x = 37 and y = 52 is a solution.