Linear systems

A linear system is a set of equations where each variable appears only to the first power. Solving it means finding values for all variables that make every equation true at once. Use elimination: add or subtract equations, or multiply an equation by a number, to remove one variable and solve for the rest. A system can have exactly one solution, no solution, or infinitely many solutions. If elimination produces a false statement like $0=5$, there is no solution. If it produces a true statement like $0=0$, there are infinitely many solutions. Otherwise, back-substitute to find the unique solution.