# [SOLVED]System of equations

#### anemone

##### MHB POTW Director
Staff member
Find all triples $(x,\,y,\,z)$ of real numbers that satisfy the system of equations

$x^3=3x-12y+50\\y^3=12y+3z-2\\z^3=27z+27x$

#### anemone

##### MHB POTW Director
Staff member
Observe the following identities:
$x^3-3x-2=(x-2)(x+1)^2\\y^3-12y-16=(y-4)(y+2)^2\\z^3-27z-54=(z-6)(z+3)^3$

Suppose $x>2$, we then have

$-12y+50=x^3-3x>2\implies y<4$

$z^3-27z=27x>54 \implies z>6$

$y^3-12y=3z-2>16 \implies y>4$

Now, assume $x<2$, we then have
$-12y+50=x^3-3x<2 \implies y>4$
$3z-2=y^3-12y>16 \implies z>6$
$27x=z^3-27z>54$ which is impossible.
THus, $x=2$ and that gives the only solution set $(x,\,y,\,z)=(2,\,4,\,6)$.