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Number of Solutions of a System of Two Linear Equations
When two linear equations in two variables are graphed on the same coordinate plane, the resulting pair of lines falls into exactly one of three categories, each determining how many solutions the system has:
- The lines intersect at one point. The single point where the lines cross is the only ordered pair that satisfies both equations, so the system has exactly one solution.
- The lines are parallel. Because parallel lines never meet, there is no point in common — the system has no solution.
- Both equations produce the same line (coincident lines). Every point on that line satisfies both equations, so the system has infinitely many solutions.
The number of solutions can also be determined without graphing by converting both equations to slope-intercept form and comparing their slopes and y-intercepts:
- Different slopes → the lines are intersecting → 1 solution.
- Same slope, different y-intercepts → the lines are parallel → no solution.
- Same slope, same y-intercept → the lines are coincident → infinitely many solutions.
This algebraic approach works because the slope and y-intercept together completely determine a line's position in the plane. If two lines differ in slope, they must cross somewhere; if they share a slope but not a y-intercept, they run in the same direction but are offset from each other and never meet; and if they share both values, they are the same line.
This three-way classification for systems of two linear equations mirrors the classification of single linear equations in one variable: a conditional equation has one solution, a contradiction has no solution, and an identity has infinitely many solutions.
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