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Solving a Linear System in Three Variables with no or Infinite Solutions - Problem 2

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!

Solving a system in three variables. So whenever we solve a system in three variables the first thing we always want to do is take it down to a system in two variables, two equations and two variables. In doing that we need to use elimination. We could use substitution, I highly recommend against it so let’s work with elimination.

Looking at this, we want to somehow get rid of a variable from pairing two equations. It doesn’t really matter which one, I see that my xs are all dealing with a little bit smaller numbers so I’m going to go ahead and get rid of x.

I have this; let’s pair the top and the middle. If I multiply the middle one by -3, I now have -3s and 3x when I add those, those cancel out. So top equation stays the same and the bottom equation we multiply by -3; -3x plus 12y minus 15z is equal to -9. We multiplied it by -3 because I knew that I wanted to add. When I add these two together what we realize is that everything is equal and opposite. My xs disappear, my ys disappear and my … I miswrote that, didn’t I? That should be a z, sorry about that. My zs disappear. So everything cancels out, even my 9 disappears and leaves us with zero is equal to zero.

What that means is that these two equations are actually representing the same plane. They are right on top of each other. So we know that these equations share the same plane, but that doesn’t tell us if we actually have a solution yet because we don’t know what’s happening with this plane.

What we have to do is we have to pair this third equation with one of the two to see if it is the same plane as well. It doesn’t matter if we do the top or the middle. 2 is a little bit smaller that 1, looking at the coefficients so let’s multiply the middle one by 2 and see what comes up.

Multiply the middle equation by 2, we end up with 2x minus 8y plus 10z is equal to 6. Bottom equation; -2x plus 8y minus 10z is equal to -6. Again we want to add. Looking at this, everything is equal and opposite so again we’re left with zero is equal to zero.

What this bottom part tells us is our bottom two equations are the same. What this top part tells us is our top two equations are the same. We can extrapolate from that, that our all three equations are actually representing the same plane. What that tells us is that these three equations intersect everywhere.

How you write this kind of depends on your teacher but all three of these equations are the same plane. So what you write is just any of these equations. Actually I’m not going to rewrite them because they are all already written over here. Any of these three equations are the same plane so you could just write that. If your teacher is into set builder notation you could also do your x, y and z such that and that insert any one of these three equations into this space because they’re all representing the same plane. Going through our elimination we determine that all three of these planes are in fact equal.

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