Systems of linear equations

A system of linear equations can be written as one matrix equation Ax = b. Row reduction (Gaussian elimination) solves systems of any size, and the determinant tells you whether there’s a unique solution.

Key facts

  • Ax = b, so x = A⁻¹b when A is invertible.
  • Gaussian elimination: use row operations to get zeros below the diagonal, then back-substitute.
  • det(A) = 0: either no solutions or infinitely many.
  • Geometrically, three equations in x, y, z are three planes.

Worked examples

Example 1

Solve x + y + z = 6, 2y + 5z = −4, 2x + 5y − z = 27.

  1. R3 − 2R1: 3y − 3z = 15, so y − z = 5.
  2. With 2y + 5z = −4: 2(z + 5) + 5z = −4 z = −2
  3. y = 3, then x = 6 − 3 + 2 = 5

Answer: x = 5, y = 3, z = −2