What is the name for a system of linear equations with the same variables?

A system of linear equations, also called a linear system, is a set of linear equations involving the same variables.

For example, 2x + y = 7 and x − y = 1 form a linear system because both equations use x and y, and each variable is raised only to the first power. Solving the system means finding values that make every equation true at the same time. In this example, the solution is x = 8/3 and y = 5/3.

Systems can have one solution, no solution, or infinitely many solutions. On a graph, two equations in two variables appear as lines: intersecting lines have one solution, parallel distinct lines have none, and overlapping lines have infinitely many.

“Simultaneous system” is sometimes used informally because the equations are solved simultaneously, but “system of linear equations” or “linear system” is the standard mathematical name. Differential systems and quadratic systems describe different kinds of equations.