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Suppose we have two linear equations in two unknowns. If the equations are plotted on a rectangular grid, the graph will fit one of these scenarios:

1) The two lines cross each other (intersect).

2) The two lines don't cross - they are parallel lines

3) The two lines fall on top of each other - they're really the same line.

In case 3) the two lines are dependent - one can be changed into the other. In cases 1) and 2) we say the lines are independent.

If the pair of equations has a solution (one or more points in common) we say they are consistent ... cases 1) and 3). In case 2) the system is inconsistent; there is no solution.

To summarize:

1) Intersecting lines are consistent and independent.

2) Parallel lines are inconsistent and independent.

3) Coincident ["happen together"] lines are consistent and dependent.

*** A second order linear system CANNOT be both dependent and inconsistent.

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Q: What is dependent and inconsistent system?
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