congratulations, all your steps were perfect, however you missed the point.

you have an equation to solve for two variables, however you only have one equation. consider:

we cannot solve for A and B as exact numbers, for instance whatever A is, we can find a value for B so that the equality is true:

let A = 2

then

2B = -A

2B = -2

B = -1

or if we let A be 7

2B = -7

B = -3.5

so there are infinitely many solutions that satisfy A + 2B = 0

likewise there are infinitely many solutions which satisfy 6A - 7C = 9

you had A = (9+7C)/6

let C = 3 then

A = (9+7*3)/6 = 30/6 = 5

so if C = 3, then A = 5

or if C = -9

A = (9 + 7*(-9))/6 = (-54)/6 = -9

so if C = -9, A = -9.

so you cannot solve for one unique solution of A and C, only a relationship between the two, so that whatever let one variable be, you can find a solution for the other one.