1.39 Linear System Definitions
Linear systems are foundational in algebra, defining equations with linear relationships and methods to solve for variables in structured ways.
Linear System Definitions establishes the vocabulary for a collection of linear equations considered together, describing what counts as a valid solution to the entire collection at once, and classifying every possible system according to how many such solutions it has—exactly one, none at all, or infinitely many.
Linear System
A linear system is a collection of two or more linear equations considered together, involving the same set of variables, with the goal of finding values for those variables that satisfy every equation in the collection simultaneously. A linear system in two variables, such as x + y = 10 paired with x − y = 2, corresponds geometrically to two lines on the coordinate plane, and solving the system means finding where, if anywhere, those lines meet.
System Solution
A system solution is a set of values, one for each variable, that satisfies every equation in a linear system simultaneously, rather than satisfying only one equation while failing another. For the system above, the system solution is x = 6 and y = 4, since substituting these values into both equations produces true statements in each case, corresponding to the single point where the two lines intersect.
Consistent System
A consistent system is a linear system that has at least one system solution, corresponding geometrically to two lines that intersect at one or more points. A consistent system with exactly one solution corresponds to two lines crossing at a single distinct point, which is the most common outcome for a system of two linear equations in two variables.
Inconsistent System
An inconsistent system is a linear system that has no system solution at all, corresponding geometrically to two parallel lines that share the same slope but different vertical-intercepts, and therefore never intersect at any point on the coordinate plane. Attempting to solve an inconsistent system algebraically produces a contradiction, such as a false numerical statement, signaling that no values exist which satisfy every equation simultaneously.
Dependent System
A dependent system is a linear system in which every equation, despite possibly being written differently, describes the exact same line, meaning every point on that line satisfies every equation in the system and the system therefore has infinitely many solutions. Attempting to solve a dependent system algebraically produces a true but uninformative statement, such as 0 = 0, signaling that the equations do not provide enough independent information to narrow the solution down to a single point.
Together, these definitions classify every linear system into one of three categories: a consistent system with one or more solutions, an inconsistent system with no solution due to parallel non-intersecting lines, or a dependent system with infinitely many solutions due to equations describing the identical line.