Solution to a System of Linear Equations

What counts as a solution. For $2x-y=5,\ x+3y=6$: solving gives the unique pair $(3,1)$, which satisfies both equations -- the two lines meet at exactly one point, and the system is called consistent with a unique solution.

For $3x+2y=5,\ 6x+4y=10$: the second equation is just the first doubled, so every solution of the first automatically solves the second. Fixing $y=t$ (any real number) gives $x=\tfrac{5-2t}{3}$ -- infinitely many solutions, one for every $t$ (the two equations describe the same line). The system is consistent with infinitely many solutions.

For $4x+y=6,\ 8x+2y=18$: substituting $y=6-4x$ from the first into the second gives the contradiction $12=18$ -- no pair $(x,y)$ satisfies both (the lines are parallel and distinct). The system is inconsistent, with no solution.

Definition 1.8. A system with at least one solution is consistent; a system with no solution is inconsistent.

Note. Interchanging two equations, scaling one equation by a non-zero constant, or replacing one equation by itself plus a non-zero multiple of another, never changes the solution set -- exactly the three elementary row operations of §1.3.1, now acting on equations instead of matrix rows.

Remark. When the number of equations equals the number of unknowns, the coefficient matrix $A$ is square. If $A$ is additionally non-singular, the system can be solved by any of three methods developed next: matrix inversion (§1.4.3.1), Cramer's rule (§1.4.3.2), or Gaussian elimination (§1.4.3.3). …