what is the fastest way to solve equations having more than two variables of 1 degree?

Question

Suppose I have four equations for four variables $$a+b+c+d=0$$ $$5a+3b+2c+6d=10$$ $$12a+21b+c+4d=30$$ $$2a+3b+4c+5d=40$$

Now what is the fastest way to find a, b, c and d?

I know of elimination and substitution out of which elimination takes lesser time. Is there any faster way than that?

I took these equations (which are written above) arbitrarily.

Answer 1

In general, unless you happen to find a particularly nice relation between the coefficients that you can exploit for a shortcut, Gaussian elimination on the augmented matrix is the way to go for manual calculations. There's a reason that is what is taught in beginning linear algebra courses!

Answer 2

I don't think there is "fastest way". What I would do is to inverse the matrix associated i.e. inverse $$\begin{pmatrix}1&1&1&1\\5&3&2&6\\12&21&1&4\\2&3&4&5\end{pmatrix}.$$ But it's quite a long work too.

Answer 3

Unless the coefficients have special patterns (e.g. $x+y=4$ and $x-y=2$, we immediately have $2x=6$ and $2y=2$ and so on), we hardly find any short cut. I'd suggest eliminating equations with small coefficients first.