Question

An important application of systems of equations arises in connection with supply and demand. As the price of the product increases, the demand for that product decreases. However, at higher prices, suppliers are willing to produce greater quantities of the product. The price at which supply and demand are equal is called the equilibrium price. The quantity supplied and demanded at that price is called the equilibrium quantity. The following models describe wages for low-level skilled labor. (demand) p = -0.307x + 5.9, (supply) p = 0.328x + 3. The equilibrium number of workers is _million. The equilibrium wage is __ dollars.

a) 10 million, $3.50
b) 12 million, $4.00
c) 15 million, $4.50
d) 18 million, $5.00

Answer 1

The equilibrium number of workers is approximately 4.57 million, and the equilibrium wage is $4.50. This is found by setting the supply and demand equations equal to each other and solving for the quantity and price at equilibrium.

Step-by-step explanation:

To find the equilibrium number of workers and the equilibrium wage, we must set the demand equation equal to the supply equation. The demand equation is p = -0.307x + 5.9, and the supply equation is p = 0.328x + 3. At equilibrium, the quantity demanded equals the quantity supplied, so we can set the two equations equal to each other to solve for 'x', which represents the number of workers in millions, and 'p', which represents the wage in dollars.

Solving the system by substitution or elimination method, we will have:

-0.307x + 5.9 = 0.328x + 3

Combining like terms yields:

0.328x + 0.307x = 5.9 - 3

0.635x = 2.9

Dividing both sides by 0.635 gives us:

x = ≈ 4.567 million

To find the equilibrium wage, substitute the value of x back into either the demand or supply equation:

p = -0.307(4.567) + 5.9 = $4.50

Therefore, the equilibrium number of workers is approximately 4.57 million and the equilibrium wage is $4.50.