# System of three equations and four variables

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• Jul 25th 2012, 04:19 AM
Wilmer
Re: System of three equations and four variables
Well, you sure seem to know what you're doing.
So if you can't solve this thing....
I gather it is quite possible that there is a "short cut" staring us in the face!
With the unwieldy equations that result in trying to solve the normal way,
it seems that there MUST be a way to greatly simplify the process, else
what teacher in his right mind would lay such a cruelty on a student?!
• Jul 25th 2012, 08:40 AM
richard1234
Re: System of three equations and four variables
If there is a real solution $\displaystyle \{x_1, x_2, x_3, x_4\}$ then $\displaystyle \frac{x_1^2 x_4^2}{4} - (x_1 - x_4)^2 \ge 0 \Rightarrow \frac{x_1^2 x_4^2}{4} \ge (x_1 - x_4)^2$. Try playing around with that inequality, as well as the other five or so you will obtain. Maybe you'll get something nice out of it...

Clearly, $\displaystyle x_1 = x_2 = x_3 = x_4 = 0$ works. But is it the only solution?
• Jul 25th 2012, 09:18 AM
studenttt
Re: System of three equations and four variables
Quote:

Originally Posted by richard1234
If there is a real solution $\displaystyle \{x_1, x_2, x_3, x_4\}$ then $\displaystyle \frac{x_1^2 x_4^2}{4} - (x_1 - x_4)^2 \ge 0 \Rightarrow \frac{x_1^2 x_4^2}{4} \ge (x_1 - x_4)^2$. Try playing around with that inequality, as well as the other five or so you will obtain. Maybe you'll get something nice out of it...

I've already tried it (built the the locus of points) but it gives nothing interesting.
• Aug 20th 2012, 06:34 AM
studenttt
Re: System of three equations and four variables
• Aug 20th 2012, 08:32 AM
MaxJasper
Re: System of three equations and four variables
Quote:

Originally Posted by studenttt
How to solve this system?
Attachment 24308
Thanks a bunch!

A simple solution is:
Let x4=1 then: x1=x2=x3=0 = trivial solution
Let x4=any number <>0 then x1=x2=x3=0 trivial solution.
Let x4=0 then x1=0.72963, x2=x3=x4=0.

Hence: the system only has solution when 3 of {x1,x2,x3,x4} be =0 at the same time!!
• Aug 20th 2012, 09:54 AM
studenttt
Re: System of three equations and four variables
Quote:

Originally Posted by MaxJasper
Hence: the system only has solution when 3 of {x1,x2,x3,x4} be =0 at the same time!!

Hence? Why? Once more: the task is to find non-trivial roots, in particular, when all variables are nonzero. So far I don't see any solution.
• Aug 20th 2012, 09:57 AM
studenttt
Re: System of three equations and four variables
Quote:

Originally Posted by MaxJasper
A simple solution is:
Let x4=1 then: x1=x2=x3=0 = trivial solution

Do you really think (0,0,0,1) is a root? We do not work with complex numbers.
• Aug 20th 2012, 10:16 AM
MaxJasper
Re: System of three equations and four variables
3 eqs 4 variables means you have infinite solutions for 3 variable while changing the 4th variable...therefore, all you get is 3 equations in terms of the 4th variable...with infinite values for 3 variables. However, this specific system has the solution I mentioned before.
• Aug 20th 2012, 10:24 AM
richard1234
Re: System of three equations and four variables
Quote:

Originally Posted by MaxJasper
A simple solution is:
Let x4=1 then: x1=x2=x3=0 = trivial solution
Let x4=any number <>0 then x1=x2=x3=0 trivial solution.
Let x4=0 then x1=0.72963, x2=x3=x4=0.

Hence: the system only has solution when 3 of {x1,x2,x3,x4} be =0 at the same time!!

Can you prove that those are the only solutions?
• Aug 20th 2012, 10:26 AM
richard1234
Re: System of three equations and four variables
Quote:

Originally Posted by studenttt
Do you really think (0,0,0,1) is a root? We do not work with complex numbers.

(0,0,0,1) is a root. Who said no complex numbers?
• Aug 20th 2012, 10:33 AM
MaxJasper
Re: System of three equations and four variables
I presented my proof by using math induction.
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