MHB System of Equations: Find Real Numbers $p,q,r,s$

[anemone]

Find all real numbers $p,\,q,\,r,\,s$ that satisfy the following system of equations:

$spq+sp+pq+qs+s+p+q=9$

$rsp+rs+sp+pr+r+s+p=9$

$qrs+qr+rs+sq+q+r+s=9$

$pqr+pq+qr+rp+p+q+r=1$

 

[kaliprasad]

Find all real numbers $p,\,q,\,r,\,s$ that satisfy the following system of equations:

$spq+sp+pq+qs+s+p+q=9$

$rsp+rs+sp+pr+r+s+p=9$

$qrs+qr+rs+sq+q+r+s=9$

$pqr+pq+qr+rp+p+q+r=1$

Spoiler

Add 1 to LHS and RHS of each expression to get
$(1+p)(1+q)(1+s) = 10$
$(1+p)(1+s)(1+r) = 10$
$(1+q)(1+r)(1+s) = 10$
$(1+p)(1+q)(1+r) = 2$

multiply all 3 and then take cube root to get
$(1+p)(1+q)(1+s)(1+r) = 10 \sqrt[3]{2}$

deviding above by 1st 3 equations
hence $(1+r)=(1+q)=(1+p) =\sqrt[3]{2}$

or $r=p=q=\sqrt[3]{2}-1$

and deviding by 4th equation we get

$s = 5\sqrt[3]{2}-1$