(1) Da=-80Pa+40Pg+400;
(2) Sa=120Pa-80;
(3) Dg=40Pa-40Pg+600;
(4) Sg=80Pg-40.
From these:
(5) Da+Dg=-40Pa+1000;
(6) Sa+Sg=120Pa+80Pg-120.
ADDITIONAL INFORMATION SUPERSEDES INITIAL SOLUTION
If D=demand, S=supply, P=price, then we're into equilibria, economically speaking.
All variables must be positive (including zero) since negative values have no meaning in this context.
Since Da≥0, -80Pa+40Pg+400≥0, 40Pg+400≥80Pa, Pg+10≥2Pa; also, since Dg≥0, 40Pa-40Pg+600≥0, 40Pa+600≥40Pg, Pa+15≥Pg.
From (5) the combined demand is Da+Dg and from (6) the combined supply is Sa+Sg. When demand and supply are in equilibrium:
-40Pa+1000=120Pa+80Pg-120, 160Pa+80Pg=1120, 2Pa+Pg=14, so Pg=14-2Pa.
We have two inequalities for Pa and Pg.
We know that Pg=14-2Pa (which is an equality, not an inequality, dictated by equilibrium conditions). When we combine this with the two inequalities they intersect when:
Pg=14-2Pa and Pa+15≥Pg and Pg+10≥2Pa; that is, when Pg+10=2Pa and Pg=14-2Pa are satisfied (and only positive values are deduced, so the other inequality is eliminated):
Pg=14-2Pa and Pg=2Pa-10, that is, 14-2Pa=2Pa-10, 4Pa=24, Pa=6, so Pg=14-12=12-10=2.
It would appear that equilibrium is reached when the two prices are Pa=6 and Pg=2.
Da=-80Pa+40Pg+400=-480+80+400=0; Dg=40Pa-40Pg+600=240-80+600=760. Combined demand=760.
Sa=120Pa-80=720-80=640; Sg=80Pg-40=160-40=120. Combined supply=640+120=760.
This may not be the correct answer. I have just applied mathematical logic, with a very limited amount of business/economics knowledge. However, I hope the answer is useful to you.
