Solving several equations at once

What is a linear system?

A linear system is several first-degree equations that must all hold at the same time. It is the most widely used tool in all of linear algebra: balancing a chemical reaction, fitting a line to measurements, computing the currents in a circuit — everything ends up as a system.

What "linear" means

An equation is linear when each unknown appears to the power 1, is not multiplied by another unknown, and does not sit inside a function.

Linear                Not linear
------                ----------
2x + 3y = 7           x^2 + y = 7      (square)
x - y + 4z = 0        xy = 5           (product of unknowns)
0.5a + b = -1         sin(x) + y = 1   (function)

A system of two equations in two unknowns is written:

{ 2x + 3y = 8
{  x -  y = -1

Solving the system means finding every pair (x ; y) that satisfies both lines at once. Here the pair (x ; y) = (1 ; 2) works: 2×1 + 3×2 = 8 and 1 - 2 = -1. The point is to find it by something other than guesswork — that is what this course is about.

Each equation is a line

This is the geometric reading, and it explains everything. In the plane, the set of points (x ; y) satisfying 2x + 3y = 8 forms a line. Solving a system of two equations therefore means looking for the points common to two lines.

Only three configurations are possible:

 Intersecting lines      Parallel lines           Identical lines
                          (distinct)
      \   /                 ______                    ______
       \ /                  ______                    ======
       / \                                          (superimposed)
      /   \

  1 common point          no common point          infinitely many points
  ONE solution            NO solution              INFINITELY MANY solutions
  (consistent system)     (inconsistent)           (consistent, undetermined)

Here is the central fact: a linear system has zero, one, or infinitely many solutions — never two, never five. This trichotomy stays true with 3, 10 or 1000 unknowns.

With three unknowns

With three unknowns, each equation ax + by + cz = d describes a plane in space. The system expresses the intersection of several planes:

Three planes meeting at a point       -> 1 solution
Three planes meeting along a line     -> infinitely many solutions
Two distinct parallel planes          -> no solution

An example from real life

Two adult tickets and three children's tickets cost €31; one adult ticket and one child's ticket cost €12. What is the price of each?

{ 2a + 3e = 31
{  a +  e = 12

The second line gives a = 12 - e. Substituting into the first: 2(12 - e) + 3e = 31, so 24 + e = 31, hence e = 7 and a = 5.

This substitution method works well on two equations. Beyond that it quickly becomes unreadable, which is why Gaussian elimination — the next lesson — is preferred.

Summary