# How to use matrix method on this system of equations

Say that I have 2x+2yz=0 2y+2xz=0 and 2z+2xy=0 how would I use matrix methods to solve this system of equations? I know you can just look at it and easily figure out what the critical points are but I want to do it the safe way. Or is using the matrix method not the easiest way here?

For extra information the crit points are (0,0,0) (-1,1,1) (1,-1,1) (1,1,-1) (-1,-1,-1)

Thanks!

Dick
Homework Helper
There is no 'matrix method' to use. It's not a system of linear equations. And they aren't really called 'critical points'. They are called 'solutions'. Just use algebra and logic. Which I think you are already doing.

There is no 'matrix method' to use. It's not a system of linear equations. And they aren't really called 'critical points'. They are called 'solutions'. Just use algebra and logic. Which I think you are already doing.

What's the "algebraic" way of doing this? Multiple rounds of substitution?

Dick
Homework Helper
What's the "algebraic" way of doing this? Multiple rounds of substitution?

Yes, basically. If e.g. x+yz=0 then either y=(-x/z) or z=0, subsitute that into the other equations. Explain how you got your solutions.

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Yes, basically. If e.g. x+yz=0 then either y=(-x/z) or z=0, subsitute that into the other equations. Explain how you got your solutions.

I just looked at this and thought of good numbers that would satisfy the equations. That's why I wanted a more systematic way of doing it. The substitution looks like a real headache to do.

Dick
Homework Helper
I just looked at this and thought of good numbers that would satisfy the equations. That's why I wanted a more systematic way of doing it. The substitution looks like a real headache to do.

OK, you just solved by inspection. Doing it systematically isn't that much of headache. Just do it. If I substitute y=(-x/z) into the second relation I get x(1-z^2)=0 after a little algebra. What does that tell you?

Well I get that x=0 and z=+-1

If I plug into the third one I get x=+-z and z=+-x. Where should I start plugging things in next?

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OK, you just solved by inspection. Doing it systematically isn't that much of headache. Just do it. If I substitute y=(-x/z) into the second relation I get x(1-z^2)=0 after a little algebra. What does that tell you?

Well I get that x=0 and z=+-1

If I plug into the third one I get x=+-z and z=+-x. Where should I start plugging things in next?

Office_Shredder
Staff Emeritus
Gold Member
Well I get that x=0 and z=+-1

That's actually an or, not an and that you want

That's actually an or, not an and that you want

Oh ok. Thank you. So now that I know x=0 or z=+-1 what should my next step be? Should I plug 1 and -1 into the first equation?

Dick