Q:

Determining $A,B,C$ in terms of $X,Y,Z$ given a linear equation for each

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 :)

A:

From the second and third equation, you get $X = B + Y$ and $Z = C + Y$. Substitute in (four times) the first to get $4A = (B + Y) + 2Y + (C + Y) = B + C + 4Y$. So $Y = (4A - B - C)/4$ and so on ($X = (4A + 3B - C)/4$ and $Z = (4A - B + 3C)/4$).

