How to solve a multiple variable linear equation

How do I solve such an eqation? (I know how to solve Mx=n using gaussian elimination but I don't know how to handle 2 variables.

Thanks

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If $x$ and $y$ are "unknown" vectors, there will almost always be infinitely many solutions. Is it that you want to find one non-trivial solution? –  André Nicolas Feb 27 '12 at 14:15
yes, I want to find a non-trivial solution that minimizes the distance between Mx and Ny –  user25843 Feb 27 '12 at 14:18
This link provides some ideas: cs.mtu.edu/~shene/COURSES/cs3621/NOTES/INT-APP/… –  Emmad Kareem Feb 27 '12 at 15:22

Basically all you need is (assuming you initialize vector y): $\vec{x}=(M^{T}M)^{-1}M^{T}(N\vec{y})$ So by finding the inverse of $M^{T}M$ and performing some matrix computations you're done.
I presume you know either $x$ or $y$. If $M$ and $N$ are large, then you should use the conjugate gradient method. Otherwise you can directly invert (or pseudo-invert) $M$ or $N$ as appropriate.