Systems of Linear Equations

A set of linear equations in the same unknowns packs into a single statement, Ax = b: the coefficients form the matrix A, the unknowns the vector x, and the right-hand sides the vector b. Solving the system means finding every x that A sends to b.

There are two ways to read it. By rows, each equation is a line (in 2D) or a plane (in 3D), and a solution is a point they all share. By columns, you are asking which combination of A's columns produces b — the column picture from earlier lessons.

A linear system has exactly one solution, no solution, or infinitely many — never, say, two. Geometrically: two lines cross at one point, are parallel and never meet, or are the same line and meet everywhere.

Which case you are in is decided by the matrix: a unique solution when A is invertible (square, non-zero determinant), and otherwise either no solution or a whole family of them.

The reliable method is elimination. Using three legal row operations — swap two rows, multiply a row by a non-zero number, add a multiple of one row to another — you clear the entries below each leading term until the matrix is upper-triangular (row echelon form). Then you back-substitute from the bottom up.

The leading entries you eliminate below are the pivots. They are the heart of the method: their number is the rank, and where they fall tells you which variables are determined and which are free.

Carry elimination further — clear above the pivots and scale them to 1 — and you reach the reduced row echelon form (RREF), where the answer can be read off directly.

A column with no pivot is a free variable: you may set it to anything, which is exactly what produces infinitely many solutions. A row that reduces to "0 = a non-zero number" is a contradiction, signalling no solution at all.