1. Problem: We have demand: $P=30-2Q$ and supply: $P=6+Q$. We want to find equilibrium price and quantity before and after tax, new supply function, tax burden division, and graph. 2. Before tax equilibrium: Set demand = supply $$30-2Q=6+Q$$ $$30-6=Q+2Q$$ $$24=3Q$$ $$Q=8$$ Find $P$: $$P=6+8=14$$ Equilibrium price $P=14$, quantity $Q=8$. 3. New supply after tax $t=3$: Tax adds to price producers receive, so effective price price consumers pay $P$ relates to producers as $$P-3=6+Q$$ Rearranged $$P=9+Q$$ New supply function: $P=9+Q$. 4. After tax equilibrium: Set demand equal to new supply $$30-2Q=9+Q$$ $$30-9=Q+2Q$$ $$21=3Q$$ $$Q=7$$ Find $P$: $$P=30-2(7)=30-14=16$$ New equilibrium price $P=16$, quantity $Q=7$. 5. Tax burden: Price rose from 14 to 16, consumers pay extra 2 units per quantity. Producers receive $P - 3 = 16 - 3 = 13$, down from 14, producers pay 1 unit. Total tax revenue: Tax rate $3$ times quantity $7$ equals $21$. 6. Summary: (a) Equilibrium before tax: $P=14$, $Q=8$. (b) New supply: $P=9+Q$. (c) New equilibrium: $P=16$, $Q=7$. (d) Consumers pay 2 units, producers pay 1 unit per item, total tax $21$. 7. Desmos equation for original and new supply and demand: Demand: $y=30-2x$ Supply before tax: $y=6+x$ Supply after tax: $y=9+x$