Systems of Linear Equations

Many Equations, One Verdict

A system of linear equations is business modeling's native language: production plans meeting capacity limits, budgets split across departments, markets clearing. In matrix form, all of it is one line:

$$Ax = b$$

Three possible fates (Rouché–Frobenius): compare the rank of $A$ with the rank of the augmented matrix $(A|b)$ and the number of unknowns $n$:

• Unique solution (compatible determinate): $rank(A) = rank(A|b) = n$. The lines/planes meet at exactly one point.
• Infinitely many (compatible indeterminate): $rank(A) = rank(A|b) < n$ — free parameters remain; the equations overlap.
• No solution (incompatible): $rank(A) < rank(A|b)$ — the equations contradict; parallel lines that never meet.

Gaussian elimination is the workhorse: use row operations (swap rows, scale a row, add a multiple of one row to another) to sweep the matrix into a triangular shape, then back-substitute from the bottom up. Each row operation preserves the solution set — you are rewriting the story, not changing its ending.

Cramer's rule is the determinant shortcut for square systems with $\det A \ne 0$:

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

where $A_i$ is $A$ with column $i$ replaced by $b$. Elegant for $2\times 2$ and $3\times 3$; Gaussian elimination wins for anything bigger.

The geometry to keep: two lines in a plane either cross once (unique), coincide (infinite), or run parallel (none). Every 1,000-variable production model ends in one of the same three fates — and last lesson's determinant is the early warning: $\det A \ne 0$ guarantees the first, happiest case.

The Rouché–Frobenius verdicts

Common pitfall: Declaring "no solution" the moment a row of zeros appears in elimination. A zero row of the form $0 = 0$ signals redundancy (infinitely many solutions); only $0 = c$ with $c \ne 0$ signals contradiction.