### How to Find Economic Equilibrium Using Algebra

The equilibrium price and quantity in a market is located at the intersection of the market supply curve and the market demand curve.

While it is helpful to see this graphically, it's also important to be able to solve mathematically for the equilibrium price P* and the equilibrium quantity Q* when given specific supply and demand curves.

### Relating Supply and Demand

The supply curve slopes upwards (since the coefficient on P in the supply curve is greater than zero) and the demand curve slopes downwards (since the coefficient on P in the demand curve is greater than zero).

In addition, we know that, in a basic market at least, the price that the consumer pays for a good is the same as the price that the producer gets to keep for the good. Therefore, the P in the supply curve has to be the same as the P in the demand curve.

The equilibrium in a market occurs where the quantity supplied in that market is equal to the quantity demanded in that market. Therefore, we can find the equilibrium by setting supply and demand equal to one another and then solving for P.

### Solving for P* and Q*

Once the supply and demand curves are substituted into the equilibrium condition, it's relatively straightforward to solve for P. This P is referred to as the market price P*, since it is the price where quantity supplied is equal to quantity demanded.

To find the market quantity Q*, simply plug the equilibrium price back into either the supply or demand equation. Note that it doesn't matter which one you use since the whole point is that they have to give you the same quantity.

### Comparison to the Graphical Solution

Since the P* and Q* represent the condition where quantity supplied and quantity demanded are the same at a given price, it is in fact the case that P* and Q* graphically represent the intersection of the supply and demand curves.

It is often helpful to compare the equilibrium that you found algebraically to the graphical solution in order to double check that no calculation errors were made.