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?
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1$\begingroup$ Your homogeneous system has the trivial solution $x = y = z = 0$. If it was unique, that would be quite boring, wouldn't it? But as @user2345215 explained, it is not unique because your equations are not linearly independent. $\endgroup$– GDumphartCommented Nov 22, 2014 at 15:36
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2 Answers
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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.
answered Nov 22, 2014 at 15:34
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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.
