I have a set of equations representing a linear transformation from the variables $A, B, C$ to $X, Y, Z$. I know that this can be solved by substitution, but I can't figure out how to carry out that substitution. How can this be done? Here are my equations:
$A = \frac{1}{4}X + \frac{1}{2}Y + \frac{1}{4}Z$
$B = X - Y$
$C = Z - Y$
Now I want to retrieve the opposite transformation. Given $A, B, C$, what are the values for $X, Y, Z$? This can be solved by inverting a matrix about them, but I am hoping for a simple approach by substitution.
For full disclosure, this is exam review for me. I know the question and answer, but how the substitution is carried out doesn't make sense to me :)