Problem 46 - Precalculus with Limits

Question

Finding the Equilibrium Point In Exercises \(45-48\) , find the equilibrium point of the demand and supply equations. $$ \begin{array}{ll}{\text { Demand }} & {\text { Supply }} \\ {p=100-0.05 x} & {p=25+0.1 x }\end{array} $$

Step-by-Step Solution

Answer

The equilibrium point is (500, 75)

Step 1: Set the two equations equal to each other

To find the equilibrium, the demand equation has to be equal to the supply equation. Hence, \(100-0.05x = 25+0.1x\).

Step 2: Move similar terms together

Take 0.05x from both sides of the equation and subtract 25 from both sides of the equation. This simplification leads to 0.15x = 75.

Step 3: Solve for x

Solve for 'x' by dividing both sides of the equation by 0.15. This leads to x = 500.

Step 4: Substitute x into one equation to find p

Substitute x=500 into the supply equation so that p = 25 + 0.1*500. This simplifies to p = 75

Step 5: Conclusion

The equilibrium point for the system of equations is (500, 75), meaning that at the price of 75, the quantity demanded and supplied will be 500 units.

Key Concepts

Demand and Supply Equations

In economics, the concepts of demand and supply form the foundation of market mechanics, and their equations are used to determine the relationship between the quantity of a good consumers are willing to buy (demand) and the quantity of the good suppliers are willing to sell (supply) at various prices.

Demand and supply equations are typically linear in introductory economics, exemplified by equations like \( p=mx+b \), where \( p \) represents price, \( x \) the quantity, and \( m \) and \( b \) are constants that represent the slope and the intercept of the line respectively. The demand equation typically has a negative slope, indicating that higher prices discourage consumer purchases, whereas the supply equation usually has a positive slope, reflecting that higher prices entice producers to sell more.

Understanding these equations is crucial for identifying how markets function and for computing the equilibrium point, which signifies the price and quantity at which the market is balanced - where the amounts supplied and demanded are equal.

Equilibrium in Economics

Equilibrium in economics represents a state where economic forces are balanced. This occurs at a specific price and quantity where the quantity demanded by consumers is exactly equal to the quantity supplied by producers. This point is critical because it determines the market price and quantity of goods that are exchanged.

To illustrate, let's consider a scenario where a product priced too high results in excess supply, as fewer consumers would be able to afford it. Conversely, if the product's price is set too low, there will be a shortage as demand exceeds supply. It's this interplay that naturally drives the price towards an equilibrium.

The equilibrium point can be graphically represented as the point where the demand and supply curves intersect. In practice, changes in market conditions, including consumer preferences and production costs, can shift these curves, altering the equilibrium point. For businesses and policymakers, understanding and predicting these shifts are crucial for making informed decisions.

Solving Linear Equations

Solving linear equations is an essential skill in algebra and forms the basis for more complex mathematical modeling. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.

The general approach to solving linear equations involves a few key steps: isolating the variable, simplifying the equation, and then solving for the variable. In the context of finding an equilibrium point, you set the demand and supply equations equal to each other and follow these steps to find the value of \( x \), which is the equilibrium quantity.

Improving the Approach

One way to enhance student understanding is to visually represent the solution process. For example, plotting both the demand and supply lines on the same graph can help students visually see the point of intersection, which corresponds to the equilibrium point. Another improvement is to use real-world examples to demonstrate how these equations can be applied, making the abstract concepts more tangible.