# Geometrically describe a linear system in two variables. 1. Geometric sense of a linear system in two variables. Describe the possible cases.

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Assuming we have two equations with two unknowns there are three possibilities; since the equations are linear they describe lines:

(1) The system could have a singular solution. The system is both consistent and the equations are independent.

**Graphically this is two lines intersecting.**

(2) The system could have no solution. The system is inconsistent.

**Graphically the lines are parallel.**

(3) The system could have infinite solutions. The system is consistent, but the equations are dependent (one equation is a multiple of the other).

**Graphically this is a single line.**

** If there is one equation in two unknowns it is a line -- not a system.

** If there are three (or more) equations in two unknowns the possibilities include:

(a) The system is consistent but the equations are dependent. One of the equations can be derived from the other two. Graphically the three lines meet at a point.

(b) The system is inconsistent. Graphically you could have three parallel lines or the lines meet at three points.

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