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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