0. Frustration

Suppose we want to solve the system of simultaneous equations

$$\begin{array}{lcr} x+2y &= &3\\ 2x+y &= &3\\ x-2y &= &1\\ 2x-y &= &2 \end{array}$$

The frustration is that there is no solution. The first two equations together, for example, have only the solution $x=1,y=1$, and the last two together have only the solution $x = 1, y = 0$. The system has more equations than variables. It is ``overdetermined''. The best we can hope for is to make the equations all approximately true, in some sense:

$$\begin{array}{lcr} x+2y &\approx &3\\ 2x+y &\approx &3\\ x-2y &\approx &1\\ 2x-y &\approx &2 \end{array}$$