[Math] Unknows solution with 3 equations and 2 unknowns

linear algebrasystems of equations

Consider a system with 3 equations and 2 unknowns that has no solutions. List all possible arrangements of the 3 equations as lines on the x-y plane.

I know for a system to have no solution the determinant must be 0. But I do not understand what possible arrangements I can have.

Best Answer

Given that equations with two variables (let's assume $x,y \in \mathbb{R}$) are graphically lines on the plane $XY$ then the arrangements are the following:

  • If there's one solution: the three lines intersect at the same point, which is the solution of the system. (The system is compatible and determined)
  • There's no solution: in this case the three lines are somehow don't intersect at the same point. So any time they don't have a point in common it will be an incompatible system.
  • They are all the same line: in that case they intersect in an infinite amount of points so there are infinite solutions to your system. (The system is compatible and undetermined).
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