Add the two equations: 2x=2p+2q+1, x=p+q+½, 1/x=1/(p+q+½)
Take x+1/x=2p and substitute for x:
p+q+½+1/(p+q+½)=2p,
Multiply through by p+q+½:
p2+2pq+q2+p+q+¼=2p2+2pq+p,
p2+q2+q+¼=2p2,
(q+½)2-p2=0,
(q+½-p)(q+½+p)=0, so p=q+½ or p=-q-½.
That is, 2p=2q+1⇒2/x=0, x→∞ or 2p+2q+1=0⇒x=0 (so 1/x→∞).
If we ignore the implications for x, we have the p/q relationships p=±(q+½).