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Classification of Systems of Two Linear Equations by Graph Type
A system of two linear equations can be classified into one of three categories based on the geometric relationship of its graphed lines, which dictates both the number of solutions and its algebraic properties:
- Intersecting Lines: The lines intersect at exactly one point, meaning the system has exactly solution. The system is classified as consistent (because a solution exists) and the equations are independent (each line has its own distinct set of solutions).
- Parallel Lines: The lines never intersect, meaning there is no solution. The system is classified as inconsistent (because no solution exists) and the equations are independent.
- Coincident Lines: Both equations represent the exact same line, meaning there are infinitely many solutions since every point on the line is a solution. The system is classified as consistent and the equations are dependent (all solutions of one equation are solutions of the other).
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Ch.4 Systems of Linear Equations - Intermediate Algebra @ OpenStax
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