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Solving a Linear System in Three Variables with a Solution - 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!

In Algebra II, sometimes we will be asked to solve systems of equations three variables. When solving these systems of equations, a 3D coordinate system is necessary since systems of equations with three variables are not linear. Therefore, solving these systems of equations by graphing is not possible. Solving by substitution would be difficult, so we often solve by addition and elimination.

Solving of a system of equations in three variables is very similar to solving a system in two variables except this time instead of dealing with lines we're actually dealing with planes that extra variable gives us a third dimension which makes a plane so what we can think about is ways that planes could intersect okay, so iImagine that the top of this box is a plane okay, so we have one plane that's just going horizontal on the bottom here and we have two other planes we're trying to figure out how these three planes can all come together okay, so one way they could all lie is if they're all on top of each other. Now there's no points where these all intersect so therefore there would be no intersection of these planes there would no solutions to this equation okay.
Another way they could all lie is if they all are sort of form a triangle, now this triangle is intersecting the bottom this triangle is intersecting the bottom and these two are intersecting as well but there's no place that all three of these are coming together so again there would be no solution to the entire system okay.
What also could happen is if this plane intersects the bottom plane so they would cross at this line right here and this third plane comes in at another angle. Now what's actually going to happen here is they all come together at this single point where this line meets this plane so we would have one point as a solution.
The last kind of scenario that could occur is again we have this plane going up and down and this plane coming in as well and they all meet at the same line so we have these two crossing at the line and the same line being here, we'd actually have a line that would be our solution so, many different ways that three planes can come together we can either have no solution, one solution or a line.
I guess the last way we could do happens from time to time is actually if all three planes are the same, similar if we had two lines that were the same line. If all these planes are the same plane, that plane itself is a solution so that's the last way that all these planes can intersect so visually that's how all these three planes can come together.

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