Find the equilibrium point of the demand and supply equations.
[tex]\begin{array}{cl}\text { Demand } & \text { Supply } \\p=110-0.05 x & p=65+0.1 x \\(x, p)=(\square) &\end{array}[/tex]
Answer (2)
Set the demand equation equal to the supply equation: 110 − 0.05 x = 65 + 0.1 x .
Solve for x : x = 0.15 45 = 300 .
Substitute x = 300 into the demand equation to find p : p = 110 − 0.05 ( 300 ) = 95 .
The equilibrium point is ( 300 , 95 ) .
Explanation
Understanding the Problem We are given the demand and supply equations: Demand: p = 110 − 0.05 x Supply: p = 65 + 0.1 x Our goal is to find the equilibrium point ( x , p ) where the demand and supply are equal. This means we need to find the quantity x and price p that satisfy both equations simultaneously.
Setting Demand Equal to Supply To find the equilibrium point, we set the demand equation equal to the supply equation: 110 − 0.05 x = 65 + 0.1 x
Solving for x Now, we solve for x :
110 − 65 = 0.1 x + 0.05 x 45 = 0.15 x x = 0.15 45 x = 300
Solving for p Now that we have the equilibrium quantity x = 300 , we can substitute this value into either the demand or supply equation to find the equilibrium price p . Let's use the demand equation: p = 110 − 0.05 x p = 110 − 0.05 ( 300 ) p = 110 − 15 p = 95
Finding the Equilibrium Point Therefore, the equilibrium point is ( x , p ) = ( 300 , 95 ) . This means that at a quantity of 300 units, the price at which consumers are willing to buy is equal to the price at which producers are willing to sell, which is $95.
Examples
Understanding supply and demand equilibrium is crucial in economics. For example, consider a local farmer's market. If the price of tomatoes is too high, demand will be low, and the farmer will have unsold tomatoes. If the price is too low, demand will be high, but the farmer might not make enough profit to justify selling them. Finding the equilibrium point helps the farmer set a price that balances supply and demand, ensuring they sell their tomatoes at a price that is both attractive to buyers and profitable for the farmer. This concept applies to virtually any market, from agricultural products to stocks and bonds.
The equilibrium point of the demand and supply equations is ( 300 , 95 ) , indicating that at a quantity of 300 units, the price is $95. This is the price at which the quantity demanded equals the quantity supplied. Thus, both consumers and producers agree on the selling price at this quantity.
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