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Solving a Linear System in Three Variables with no or Infinite Solutions - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Sometimes we have a system of equations that has either infinite or zero solutions. We call these no solution systems of equations. When we solve a system of equations and arrive at a false statement, it tells us that the equations do not intersect at a common point. One scenario is that 2 or more of the planes are parallel or that two of the planes intersect and the other intersects at a different point.

Solving a system in three variables. Whenever we're dealing with equation with three variables that means we're dealing with an actual dimension. In this case everything is a first degree so we are dealing with a plane so we actually have here is three planes and we need to figure out how three planes can intersect okay? The most common way that we can find them intersecting is the if they actually intersect at a point so if this bottom plane is this bottom surface of the plane we have a plane that comes into that and a plane that comes in as well, they all meet at this point right down here okay?
What we're going to talk about now is what happens when they don't meet at a single point. We're going to solve these out mathematically but what we want to do is figure out how we can interpret our results okay, and there's a couple of things that can happen where they don't intersect at one point. Again if this bottom surface is a plane, we can have parallel planes in which case we have no intersection for all three or they can all be the same plane altogether in which case they intersect everywhere okay, so what we're going to do is do these algebraically and see when this weirdness happens, what that means physically is happening with our planes.

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