Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the point of equilibrium of the demand and supply equations. The point of equilibrium is the price \(p\) and the number of units \(x\) that satisfy both the demand and supply equations. Demand \(\quad\) Supply \(p=500-0.4 x \quad p=380+0.1 x\)

Step-by-Step Solution

Answer

The equilibrium point is at (240, 404), meaning 240 units will be bought or sold at an equilibrium price of 404.

Step 1: Set the two equations equal to each other

500 - 0.4x = 380 + 0.1x, this is because at the point of equilibrium, the price consumers are willing to pay is equal to the price producers are willing to sell at.

Step 2: Solve for x

To solve for x, collect like terms and isolate x. This is achieved as: 0.4x + 0.1x = 500 - 380; hence 0.5x = 120. Dividing both sides by 0.5 gives x = 240. Thus, the number of units that will be bought and sold at equilibrium is 240.

Step 3: Substitute x in either the supply or demand equation to find p

Substitute x = 240 into the equation of demand, we get: p = 500 - 0.4(240). This gives: p = 404. Therefore,the price at which these units will be bought or sold (in equilibrium) is 404.

Key Concepts

Understanding the Demand Equation

The demand equation is a mathematical representation of the relationship between the price of a good and the quantity demanded by consumers. In this equation, the price ( \( p \) ) depends on the number of units ( \( x \) ). Typically, as the price decreases, the demand or quantity of goods consumers want to buy increases. This inverse relationship is depicted in our demand equation: \( p = 500 - 0.4x \).

When examining the demand equation, it's important to notice its slope. The slope indicates the rate at which demand changes with price. Here, the slope is \(-0.4\), showing that for every additional unit ( \( x \) ), the price \( p \) decreases by 0.4 units. Understanding this equation helps us predict consumer behavior as prices fluctuate.

Breaking Down the Supply Equation

The supply equation shows how the quantity supplied by producers is related to the market price. It functions in contrast to the demand equation, usually with a positive slope. Here, the supply equation is given by \( p = 380 + 0.1x \).

This implies that as the number of units ( \( x \) ) increases, so does the price ( \( p \) ) of the good, reflecting that producers demand higher prices to supply more goods. The slope in this equation is \(0.1\), indicating that for every additional unit supplied, the price increases by 0.1 units. Understanding this helps one to analyze how much producers are willing to sell at varying price levels.

Solving Equations for Equilibrium

Solving the demand and supply equations is crucial to find the economic equilibrium. To do this, you need to set both equations equal because the equilibrium point is where the quantity demanded equals the quantity supplied at a certain price.

In this exercise, the demand equation is \( p = 500 - 0.4x \) and the supply equation is \( p = 380 + 0.1x \). By setting \( 500 - 0.4x = 380 + 0.1x \), you simplify and solve for \( x \).

• First, collect all terms involving \( x \) on one side: \( 0.4x + 0.1x = 500 - 380 \).
• This simplifies to \( 0.5x = 120 \).
• Solving for \( x \), you divide by 0.5 to find \( x = 240 \).

Thus, 240 units is the equilibrium quantity where supply meets demand.

Exploring Economic Equilibrium

Economic equilibrium is a significant concept in economics, referring to the state where the quantity demanded equals the quantity supplied. This balance ensures that there is no excess supply or demand in the market. At this point, the market is efficient, and resources are optimally allocated.

To find the equilibrium price from our exercise, substitute \( x = 240 \) back into either equation. Using the demand equation: \( p = 500 - 0.4 imes 240 \), we find \( p = 404 \). This equilibrium price of 404 means both consumers and producers agree on the price at which the 240 units will be traded. At economic equilibrium, the market operates smoothly without any unintended pressure on prices.