Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 :)

share|improve this question
add comment

2 Answers

up vote 2 down vote accepted

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 $Z = (4A - B + 3C)/4$).

share|improve this answer
add comment

So, you have the following matrix equation $A=PX$:

$$\begin{bmatrix}A\\B\\C \end{bmatrix}=\begin{bmatrix}\frac 1 4 & \frac 1 4 &\frac 1 4\\ 1&0&-1&\\0&-1&1\end{bmatrix}\begin{bmatrix}X\\Y\\Z\end{bmatrix}$$

Yo're looking for a $Q$ such that $X=QA$. Working from what you have, you must see that $Q=P^{-1}$.

Hope you can evaluate the inverse...


Hint

Invert the co-efficient matrix...

share|improve this answer
    
Sorry. I pointed out in the question that I know this can be solved by inverting a matrix, but I'm looking for the substitution approach :) –  Deod Apr 17 '12 at 17:37
    
I am sorry, I did not read the question properly. :-( –  user21436 Apr 17 '12 at 17:44
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.