Systems of Linear Equations

Many Equations, One Verdict

A system of linear equations is business modeling's native language, written compactly as $Ax = b$. Three fates govern every system, compared by the rank of $A$, the augmented matrix $(A|b)$, and unknowns $n$.

Common pitfall: A zero row like $0 = 0$ means redundancy (infinite solutions), while $0 = c$ where $c \ne 0$ means no solution.

Solving Methods

Gaussian elimination uses row operations (swapping, scaling, adding rows) to transform the matrix into triangular shape, then back-substitutes from the bottom up. Row operations preserve the exact solution set.

Cramer's rule solves square systems when $\det A \ne 0$ via the formula:

$$x_i = \frac{\det A_i}{\det A}$$

Here, $A_i$ is $A$ with column $i$ replaced by $b$. It shines for $2 \times 2$ and $3 \times 3$ systems, but Gaussian elimination wins for anything larger.

The geometry to keep: last lesson's determinant is your early warning. $\det A \ne 0$ guarantees the unique solution case immediately.