Our study of Gaussian elimination and the row canonical form gives us a method to solve systems of linear equations

First of all, it is more confortable to work with the associated matrix of the system

To this matrix we perform elementary operations until we reach its row reduced form. We know that elementary operations don't modify the set of solutions, that's the crucial point!

The last rows are the simpler ones: that's why we have to analyze the new system from the last row upwards

1. A row full with zeros

represents the equation

which essentially does nothing. So we simply skip such rows

2. A row with a pivot at the very end

represents the equation

which is impossible! If we ever encounter such a row, our system is incompatible

3. If we don't encounter such impossible rows, our system will be compatible. To find out why, keep in mind that every other row above has a pivot. If there are as many pivots as unknowns, then our reduced canonical form (after removing zero rows) will be something like

for some altered coefficients $b_i'$, which represents the system

that obviously has just one solution

and thus is determinate

If there are more unknowns than pivots, the unknowns whose column has no pivot will be free: we may choose the value we want for them. Once these values are fixed, the unknowns associated to a pivot may be solved in terms of the free ones. Since we can choose, there are many solutions, and the system is indeterminate. For instance, the matrix

would represent the system

$x_2$, $x_5$ and $x_6$ are free and they may be assigned any value in $\mathbb{K}$; the other variables are solved in terms of these

The compatibility and determination of a system $(A|b)$ are found via the number of nonzero rows in the row canonical form of $A$ and $(A|b)$ - this number has a special name, the rank, and the discussion has a special name too, the Rouché-Capelli Theorem