To find the equilibrium in a market, we utilize the equations for supply and demand curves. The first step involves ensuring that the same variable is isolated in both equations, whether it be price or quantity. This means that we want either price or quantity to be on one side of both equations. Once the variables are aligned, we can proceed to the next step.

The second step is to set the demand and supply equations equal to each other. At equilibrium, the quantity supplied equals the quantity demanded, which allows us to equate the two equations. For example, if we have the demand equation \( P = 800 - 2Q_D \) and the supply equation \( P = 200 + Q_S \), we can set them equal: \( 800 - 2Q = 200 + Q \). Here, we can simplify our notation by using \( Q \) to represent both \( Q_D \) and \( Q_S \) since they are equal at equilibrium.

In the third step, we solve for the variable \( Q \). Rearranging the equation \( 800 - 2Q = 200 + Q \) leads to \( 800 - 200 = 3Q \), which simplifies to \( 600 = 3Q \). Dividing both sides by 3 gives us the equilibrium quantity \( Q^* = 200 \).

Next, we substitute this equilibrium quantity back into one of the original equations to find the equilibrium price. Using the supply equation \( P = 200 + Q \), we substitute \( Q = 200 \) to find \( P^* = 200 + 200 = 400 \). Thus, the equilibrium price is \( P^* = 400 \) and the equilibrium quantity is \( Q^* = 200 \).

To confirm our results, we can also solve using the other variable. If we isolate quantity in the supply equation and set it equal to the demand equation, we can derive the same equilibrium price and quantity. For instance, using the demand equation \( Q_D = 400 - \frac{1}{2}P \) and substituting \( P = 400 \) yields \( Q_D = 400 - \frac{1}{2}(400) = 200 \), confirming our earlier findings.

This process illustrates the importance of algebraic manipulation in finding market equilibrium, demonstrating that regardless of which variable is isolated, the results will remain consistent. Understanding these steps is crucial for analyzing market dynamics and predicting outcomes based on changes in supply and demand.

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