# Thread: Proof In a System of Equations

1. ## Proof In a System of Equations

Prove that x=y=z in the equations

$x+y^2+z^4=0$
$y+z^2+x^4=0$
$z+x^2+y^4=0$

where x, y, and z are real numbers.

2. Originally Posted by Winding Function
Prove that x=y=z in the equations

$x+y^2+z^4=0$
$y+z^2+x^4=0$
$z+x^2+y^4=0$

where x, y, and z are real numbers.
Can you give us some indication as where this problem comes from?

3. Originally Posted by danny arrigo
Can you give us some indication as where this problem comes from?
An Alg2/Trig Text. Is it too easy/difficult?

4. Originally Posted by Winding Function
Prove that x=y=z in the equations

$x+y^2+z^4=0$
$y+z^2+x^4=0$
$z+x^2+y^4=0$

where x, y, and z are real numbers.
I'm sorry, are you saying prove that $x=y=z$ is the only solution?

5. ## Dummy bro = plz read books

x=-(y^2+z^4)

y=-(z^2+x^4)

z=-(x^2+y^4)

let us consider that x=y=z and let us assume a real value which is positive.
clearly any real number raised to even powers like 2 or 4 must give positive values
as seen from the equation a positive value cannot be equal to a negative value. so the question is wrong.

now let us assume a real negative value.
x=y=z=-a
clearly LHS of the equations are negative.
-a raised to powers of 2 or 4 will produce positive values which are much larger or smaller (in case of negative fractions) than -a; then
due to the negative sign outside bracket RHS become negative values which are much smaller or larger (in case of negative fractions) than -a

Hence we draw conclusions that the equations are wrong.

6. Originally Posted by susrut
x=-(y^2+z^4)

y=-(z^2+x^4)

z=-(x^2+y^4)

[...] now let us assume a real negative value.
x=y=z=-a
clearly LHS of the equations are negative.
-a raised to powers of 2 or 4 will produce positive values which are much larger or smaller (in case of negative fractions) than -a; then
due to the negative sign outside bracket RHS become negative values which are much smaller or larger (in case of negative fractions) than -a

HENCE WE DRAW CONCLUSIONS THAT THE EQUATIONS ARE WRONG!!!!!!
The equation $-a=-(a^2+a^4)\Longleftrightarrow a(1-a-a^3)=0$ has at least one nontrivial solution because the polynomial $1-X-X^3$ has one real root... The equations are not wrong.

7. susrut,

It's very rude to use a title such as "Dummy bro = plz read books".

8. Originally Posted by Mathstud28
I'm sorry, are you saying prove that $x=y=z$ is the only solution?
I'm trying to prove that for any set of solutions, (x,y,z), x=y=z.

9. HELLO FLYINGSQUIRREL

first of all i thank you

1-X-X^3=0 has one real root which comes to be NEARLY equal to 0.68
so x=y=z~0.68 is a solution to the equations
but it is not a strong enough proof as it is a backdoor approach

I did not meant to prove that $x=y=z$, I just wanted to show that the three equations we are given are correct.