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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: 1. The lines intersect at one point, meaning the system has exactly one solution. 2. The lines are parallel and never meet, meaning the system has no solution. 3. Both equations produce the same line (coincident lines), meaning the system has infinitely many solutions. The number of solutions can also be determined algebraically without graphing by converting both equations to slope-intercept form and comparing their slopes and y-intercepts.
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