Gaussian Elimination Method

Gaussian elimination method row-reduces the augmented matrix $[A\,|\,B]$ to row-echelon form, then reads the unknowns off from the bottom equation upward -- the method of back substitution. Unlike matrix inversion and Cramer's rule, this method works even when $A$ is singular or rectangular (any number of equations, any number of unknowns), which makes it the most broadly applicable of the three.

Worked illustration. Solve $x+2y+z=6,\ 2x-y+z=1,\ 3x+y+2z=11$. $[A|B]=\begin{pmatrix}1&2&1&|&6\\2&-1&1&|&1\\3&1&2&|&11\end{pmatrix}\xrightarrow{R_2\to R_2-2R_1,\ R_3\to R_3-3R_1}\begin{pmatrix}1&2&1&|&6\\0&-5&-1&|&-11\\0&-5&-1&|&-7\end{pmatrix}\xrightarrow{R_3\to R_3-R_2}\begin{pmatrix}1&2&1&|&6\\0&-5&-1&|&-11\\0&0&0&|&4\end{pmatrix}$. The last row reads $0=4$, a contradiction -- this particular system is inconsistent (illustrating that echelon form also detects inconsistency, not just unique solutions).

The general shape when a unique solution exists. Echelon form $\begin{pmatrix}1&*&*&|&*\\0&1&*&|&*\\0&0&1&|&*\end{pmatrix}$ gives the bottom equation directly (e.g. $199z=398\Rightarrow z=2$), substitutes upward into the middle equation to find $y$, then upward again into the top equation to find $x$ -- the method of back substitution.

Word problems. Fitting $v(t)=at^2+bt+c$ to three data points, splitting an investment across interest-bearing bonds, or determining a polynomial's coefficients from Remainder-Theorem data, all give a linear system whose augmented matrix is row-reduced exactly as above. …