TITLE: Solution of system of three variables
QUESTION [5 upvotes]: On solving 
$$ 2x - 4y + z = 0 $$ 
$$ x + y - 4z = 0 $$ 
$$ x - y - z = 0 $$ 
I get 
$$ y = 0.6 x $$
$$ z = 0.4 x$$
 I thought that there was a rule of thumb that you need as many independent equations as the number of unknowns. So, why don't I get a unique solution in this case?

REPLY [5 votes]: Because
$$\begin{pmatrix}2&-4&1\\1&1&-4\\1&-1&-1\end{pmatrix}\sim\begin{pmatrix}0&-2&3\\0&2&-3\\1&-1&-1\end{pmatrix}\sim\begin{pmatrix}0&0&0\\0&2&-3\\1&-1&-1\end{pmatrix},$$
so the system isn't independent.

REPLY [3 votes]: Hint: The matrix of the coefficients of the given system of equations is $$A=\begin{pmatrix}2&-4&1\\1&1&-4\\1&-1&-1\end{pmatrix}$$ with $det(A)=0$ so the equations are not independent. Therefore the fact that you do not get a unique solution is not a  contradiction to what you (correctly) know.