The corner points of a bounded feasible region determined by the following system of linear inequalities x + 3y ≤ 60, x + y ≥ 10, x ≤ y, x ≥ 0, y ≥ 0…

The maximum value of $z$z occurs at two corner points: $(15, 15)$(15,15) and $(0, 20)$(0,20).

In linear programming, when the maximum occurs at two corner points, both points give the same maximum value of $z$z and the objective function line is parallel to the edge joining these two points.

---

At point $(15, 15)$(15,15):

$$z = 2p(15) + q(15)$$z=2p(15)+q(15)

$$z = 30p + 15q$$z=30p+15q

At point $(0, 20)$(0,20):

$$z = 2p(0) + q(20)$$z=2p(0)+q(20)

$$z = 20q$$z=20q

---

Since both points give the maximum value:

$$30p + 15q = 20q$$30p+15q=20q

$$30p = 20q - 15q$$30p=20q−15q

$$30p = 5q$$30p=5q

$$6p = q$$6p=q

---

Therefore, the relation between $p$p and $q$q is $q = 6p$q=6p.