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Systems of Linear Equations

A pile of linear equations is really one matrix equation, A*x = b, hiding in plain sight. Once you see that, the geometry follows: each equation is a line (or a plane), and solving means finding where they all meet — which can happen in exactly one place, nowhere, or infinitely many places.

From equations to A*x = b

A system of linear equations is a set of equations in the same unknowns, where each unknown appears only multiplied by a number and added — no squares, no products of unknowns. Take this pair in x and y.

  2x + 1y = 5
  1x + 3y = 6

packs into  A*x = b  with
  A = [[2,1],[1,3]]   x = (x,y)   b = (5,6)
The coefficients become A, the unknowns become x, the right side becomes b.

Read A*x with last guide's column view: we are asking what weights x make a blend of A's columns equal b. A whole system collapses into one tidy line: **A*x = b**.

The geometry: where lines meet

Each equation in two unknowns draws a line in the plane. A solution is a point (x, y) that sits on every line at once. With three unknowns, each equation is a plane in space, and a solution is a point shared by all the planes.

One, none, or infinitely many

Two lines can relate in exactly three ways, and so can the system: they cross at one point (one solution), they are parallel and never touch (no solution), or they are the same line (infinitely many solutions). There is no fourth case — a linear system never has exactly two or exactly five solutions.