Matrices & linear systems

system of linear equations

A system of linear equations is several straight-line equations that must all hold at once, sharing the same unknowns. For example: x + y = 5 and x - y = 1. A solution is a set of values that satisfies every equation together — here x=3, y=2.

This is the central problem of linear algebra. The whole system packs neatly into a single matrix equation A*x = b, where A holds the coefficients, x is the vector of unknowns, and b is the vector of right-hand sides.

Such a system can have exactly one solution (the lines cross at one point), no solution at all (parallel lines that never meet), or infinitely many (the same line written twice). Figuring out which case you are in, and finding the answer, is what methods like elimination are for.

x + y = 5, x - y = 1 <=> [[1,1],[1,-1]] * (x, y) = (5, 1)

Two equations packed as A*x = b; solution x=3, y=2.

Linear means the unknowns appear only to the first power — no squares, no products of unknowns.

Also called

linear systemsimultaneous equations线性方程组線性方程組联立方程