Systems of linear equations

Intuitive problem

At a market stall two baskets of mixed fruit are priced but not itemized. The first holds three apples and two oranges for a known total, the second one apple and four oranges for another known total. What is each fruit worth? One basket alone cannot say: three apples and two oranges summing to a price is consistent with many apple-and-orange prices, a whole line of them. The second basket supplies a second constraint, and where the two constraints agree there is a single answer. The general question is when several constraints on the same unknowns pin down one state, and how to find it without guessing.

Definition

A linear equation constrains its unknowns through sums of constant multiples, with no unknown multiplied by another and none raised to a power. A system is a collection of such equations required to hold at once. A solution is an assignment of values to the unknowns that satisfies every equation simultaneously, and the solution set is the collection of all such assignments. The method's central claim is that this set can only take three forms: a single point, the empty set, or an infinite family. Nothing else is possible for linear equations, and part of the work is seeing why.

Derivation

Elimination is the engine, and its legitimacy is the point worth deriving. The permitted move is to replace one equation with itself plus a multiple of another, leaving the rest untouched. This move is reversible: having added a multiple of the second equation to the first, subtracting the same multiple restores the original, so no information is gained or lost and the solution set is exactly preserved. Because both the forward and backward moves are legal, the transformed system allows precisely the assignments the original did.

Contrast this with the tempting but wrong move of summing all the equations into one. That is irreversible: from the sum alone the originals cannot be recovered, and a constraint has been dissolved, so the solution set enlarges. This is the same discipline the entry on equations as constraints applies to extraneous solutions, here at the level of whole systems. Elimination works by repeatedly using the reversible move to remove one unknown at a time until a single equation in a single unknown remains, which is solved directly and then substituted back.

Visual representation

Two unknowns give a picture. Each linear equation is a line in the plane. Two lines can cross at one point, which is the unique solution; they can be parallel and never meet, which is no solution; or they can be the same line, which is infinitely many solutions. These three pictures exhaust the possibilities, and they are the geometric face of the three-way classification. The exhaustiveness is a fact about linearity: straight lines admit no other arrangement. Replace a line with a curve, as in x squared equals y, and the trichotomy breaks, which is why the guarantee belongs to linear systems specifically.

Counterexample

Consider two constraints that look independent but are not: x plus y equals 3, and 2x plus 2y equals 7. The second, halved, says x plus y equals 3.5, contradicting the first. The lines are parallel, the system is inconsistent, and no assignment satisfies both, a consistent-looking system with an empty solution set. A near miss is instructive too: if the second total were only slightly off from twice the first, the lines would be nearly parallel and cross at a point enormously sensitive to the exact numbers, which is the entryway to conditioning treated elsewhere. A system can also be overdetermined, with more equations than unknowns, in which case it usually has no exact solution unless the extra equations happen to agree.

Common mistakes

The most damaging error is subtracting one equation from another and then discarding one of the originals, which silently drops a constraint and enlarges the solution set. Elimination keeps every equation; it only replaces one with a combination, never deletes. A second error is reading a single satisfied equation as a solved system when other equations remain unchecked. A third is assuming a square system, as many equations as unknowns, always has a unique solution; it does only when the equations are genuinely independent.

Application

Systems model any situation where several balances hold at once. In a small network of flows, conservation at each junction, that what enters equals what leaves, gives one linear equation per node, and solving the system recovers the unknown flows. Mixture problems, where fixed ingredients combine to hit target totals, take the same form. The method is old: the fangcheng chapter of the Nine Chapters on the Mathematical Art set out this elimination on a counting board in Han-dynasty China, close to two thousand years before Carl Friedrich Gauss's name became attached to it in Europe.

Practice problems

Work these and check each by substitution. First, solve the market stall above: three apples and two oranges cost 13 units, one apple and four oranges cost 11 units; find each price and verify both totals. Second, take a fresh flow network in a new setting: water enters a junction at 10 liters per second and splits into two pipes, one of which is measured carrying 4 liters per second more than the other; write the conservation equation at the junction and the measurement relation, solve for both pipe flows, and confirm they sum to the inflow. The success criterion is not a remembered procedure but a solution that survives substitution back into every original equation.