12.3 Back substitution

To solve an echelon system resulting from Gaussian elimination, start with the last equation and work backwards to the first. At any stage,

• If all the coefficients of the unknowns are zero, and the right hand side too, the equation is trivial. Ignore it.
• If all the coefficients of the unknowns are zero, and the right hand side is not, there is no way to satisfy it. So there is no solution to the given system of equations. Note so explicitly.
• In all other cases, solve the equation for the pivot variable (the variable with the pivot, i.e. with the first nonzero coefficient.) Take the other terms to the right hand side and substitute in anything you already learned in solving the previous equations below it.