To analyze the impact of a tax on equilibrium price and quantity, we can follow a systematic approach using algebra. When a tax is imposed on suppliers, it effectively shifts the supply curve. For instance, if suppliers are taxed $1 per unit, we need to adjust the supply equation accordingly.

Consider the original supply equation given by Q_s = 2P - 6 and the demand equation Q_d = 10 - P. To account for the tax, we replace the price P in the supply equation with P - tax, leading to the modified supply equation:

Q_s = 2(P - 1) - 6

Expanding this gives:

Q_s = 2P - 2 - 6 = 2P - 8

Next, we find the new equilibrium by setting the modified supply equal to the demand:

2P - 8 = 10 - P

Solving for P, we combine like terms:

3P = 18

Thus, the new equilibrium price P is:

P = 6

To find the equilibrium quantity, we substitute P back into the demand equation:

Q_d = 10 - 6 = 4

At this point, we have determined that the equilibrium price is $6 and the equilibrium quantity is 4 units. However, it is essential to distinguish between the price paid by buyers and the price received by sellers. The price paid by buyers, denoted as P_B, is the equilibrium price of $6. Conversely, the price received by sellers, denoted as P_S, is calculated by subtracting the tax from the buyer's price:

P_S = P_B - tax = 6 - 1 = 5

In summary, after the tax is applied, buyers pay $6, sellers receive $5, and the equilibrium quantity remains at 4 units. This analysis illustrates how taxes can affect market dynamics, shifting the burden between buyers and sellers while maintaining the overall market equilibrium.