# Thread: System of three equations and four variables

1. ## 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?!

2. ## Re: System of three equations and four variables

If there is a real solution $\{x_1, x_2, x_3, x_4\}$ then $\frac{x_1^2 x_4^2}{4} - (x_1 - x_4)^2 \ge 0 \Rightarrow \frac{x_1^2 x_4^2}{4} \ge (x_1 - x_4)^2$. 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, $x_1 = x_2 = x_3 = x_4 = 0$ works. But is it the only solution?

3. ## Re: System of three equations and four variables

Originally Posted by richard1234
If there is a real solution $\{x_1, x_2, x_3, x_4\}$ then $\frac{x_1^2 x_4^2}{4} - (x_1 - x_4)^2 \ge 0 \Rightarrow \frac{x_1^2 x_4^2}{4} \ge (x_1 - x_4)^2$. 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.

4. ## Re: System of three equations and four variables

Any new ideas, please?

5. ## Re: System of three equations and four variables

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!!

6. ## Re: System of three equations and four variables

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.

7. ## Re: System of three equations and four variables

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.

8. ## 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.

9. ## Re: System of three equations and four variables

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?

10. ## Re: System of three equations and four variables

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?

11. ## Re: System of three equations and four variables

I presented my proof by using math induction.

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