Question

Supply and Demand The supply and demand equations for a small LCD television are given by \(\left\\{\begin{array}{ll}p+0.53 x=1542 & \text { Demand } \\\ p-0.37 x=300 & \text { Supply }\end{array}\right.\) where \(p\) is the price (in dollars) and \(x\) represents the number of televisions. For how many units will the quantity demanded equal the quantity supplied? What price corresponds to this value?

Short answer

Solving the equation in Step 2 gives us the number of units for which the quantity demanded equals the quantity supplied (x). Substituting 'x' into the equation in Step 3, gives us the corresponding price (p).

Step-by-step solution

Reformat the equations in standard form

First, rewrite the equations in a more friendly format. By moving terms around, we can get the equations in the form of 'p = ...'. This makes it easier to compare.From \(p + 0.53x = 1542\) (Demand), we get \(p = 1542 - 0.53x\)From \(p - 0.37 = 300\) (Supply), we get \(p = 300 + 0.37x\)

Find the number of televisions where demand equals supply

Next, we can determine the quantity of televisions. Set the two 'p' equations equal to each other and then solve for 'x':\[1542 - 0.53x = 300 + 0.37x\]Combine like terms to isolate 'x' and solve to find 'x':\[(0.53 + 0.37)x = 1542 - 300 \]\[x = \frac{(1542 - 300)}{(0.53 + 0.37)}\]

Determine the price at which demand is equal to supply

Once we have found the value of 'x', we can determine the price 'p'. We can substitute the found 'x' in one of our original 'p' equations. May use any, but let's use the Demand one. \[p = 1542 - 0.53x\]Substitute the value of 'x' from Step 2 into this equation to find 'p'. This will give us the price where the quantity demanded equals the quantity supplied.

Key concepts

Equations in standard form

In the world of economics, equations in standard form help us express relationships in an organized way. For the supply and demand of a product, the equations often include variables representing price and quantity. Standard form can be given as: \(ax + by = c\), where \(x\) and \(y\) symbolize variables like quantity and price.

In our exercise, we start with the demand equation \(p + 0.53x = 1542\) and the supply equation \(p - 0.37x = 300\). To simplify comparison, it's useful to express \(p\) in terms of \(x\) or \(x\) in terms of \(p\). We reformulate these to:

• Demand: \(p = 1542 - 0.53x\)
• Supply: \(p = 300 + 0.37x\)

Rewriting these equations helps in aligning them for further analysis, especially when determining where demand equals supply.

Quantity demanded

Quantity demanded refers to the amount of a product that consumers are willing and able to purchase at a given price. It's represented by the variable \(x\) in our equation. As prices change, the quantity demanded often changes as well.

For the given demand equation \(p = 1542 - 0.53x\), \(x\) shows how many LCD televisions consumers want at different price points. The negative sign before the coefficient 0.53 indicates that as the price \(p\) increases, the quantity demanded \(x\) decreases. This inverse relationship is a common characteristic in economic demand curves, highlighting consumer behavior.

Quantity supplied

Quantity supplied showcases how much of a product producers are willing and able to sell at a specific price. Analogous to demand, it's also represented by \(x\) in the supply equation.

In this context, the supply equation is \(p = 300 + 0.37x\). The positive sign before 0.37 signifies that as the price \(p\) rises, so does the quantity supplied \(x\). This direct relationship illustrates that higher prices generally encourage producers to supply more because they seek to maximize profits. Understanding this behavior aids in analyzing how markets reach equilibrium.

Price determination

Price determination is the process of establishing the market price where the quantity of a product demanded equals the quantity supplied. This balance point is known as market equilibrium.

Our task involves finding this price when \(x\), the quantity demanded, matches the quantity supplied. By solving the previously formatted equations \(1542 - 0.53x = 300 + 0.37x\), we first determine \(x\). Then, substituting this \(x\) back into either the demand or supply equation, we find the corresponding price \(p\).

• This value represents the market equilibrium price.
• It ensures no surplus or shortage of televisions in this context.

Achieving this equilibrium allows for optimal allocation of resources in an economy.