In economics, the concept of equilibrium is crucial for understanding how supply and demand interact in a market. Equilibrium occurs at the point where the supply curve and the demand curve intersect on a graph. This intersection indicates the price at which the quantity demanded by consumers equals the quantity supplied by producers.

To find this equilibrium point, we can use specific equations for quantity demanded and quantity supplied. For instance, if the quantity demanded is represented by the equation \( Q_d = 400 - \frac{1}{2}P \) and the quantity supplied by \( Q_s = P - 200 \), we can determine the equilibrium by selecting various price points and calculating the corresponding quantities.

Starting with a price of 0, we find that the quantity demanded is 400, while the quantity supplied is -200, which is not meaningful in a practical context. Therefore, we need to choose a higher price. For example, at a price of 400, both the quantity demanded and quantity supplied equal 200, indicating that we have found an equilibrium point. This is significant because it shows that at this price, the market is balanced.

Continuing with a price of 600, we calculate the quantity demanded as \( Q_d = 400 - \frac{1}{2} \times 600 = 100 \) and the quantity supplied as \( Q_s = 600 - 200 = 400 \). This further illustrates how changes in price affect the quantities demanded and supplied.

When plotting these points on a graph, we can visualize the demand curve (which slopes downward) and the supply curve (which slopes upward). The intersection of these two curves confirms our earlier calculations, showing that the equilibrium price is 400 and the equilibrium quantity is 200. This graphical representation reinforces the algebraic findings and provides a clear understanding of market dynamics.

In summary, the equilibrium point is where the market clears, meaning there is no surplus or shortage. Understanding how to derive and graph these equations is essential for analyzing market behavior and making informed economic decisions.