College Algebra 9th txtbk

436 CHAPTER 7 SYSTEMS OF EQUATIONS AND MATRICES
Price per pound ($)
5
3
p
Z Figure 6
Supply
Equilibrium
point
(13, 3)
SOLUTION
Demand
13 20
Thousands of pounds
q
To find the equilibrium quantity, we solve the linear system
p 0.2q 5.6 Demand equation (consumer)
p 0.1q 1.7 Supply equation (supplier)
using substitution (substituting p 0.2q 5.6 into the second equation).
p 0.1q 1.7
0.2q 5.6 0.1q 1.7
5.6 0.3q 1.7
3.9 0.3q
q 13 thousand pounds
Substitute p 0.2q 5.6.
Add 0.2q to both sides.
Subtract 1.7 from both sides.
Divide both sides by 0.3.
Equilibrium quantity
Now substitute q 13 back into either of the original equations in the system and solve
for p (we choose the second equation):
p 0.1(13) 1.7
p $3 per pound Equilibrium price
So if the price of cherries is $3 per pound, then the supplier would supply 13,000 pounds
of cherries and the consumer would demand (purchase) 13,000 pounds of cherries. In other
words, the market would be in equilibrium (see Fig. 6).
MATCHED PROBLEM 9
The price–demand and price–supply equations for strawberries in a certain city are
p 0.2q 4
p 0.04q 1.84
Demand equation
Supply equation
where q represents the quantity in thousands of pounds and p represents the price in dollars.
Find the equilibrium quantity and the equilibrium price.
EXAMPLE 10 Production Scheduling
A garment industry manufactures three shirt styles. Each style shirt requires the services of
three departments as listed in the table. The cutting, sewing, and packaging departments have
available a maximum of 1,160, l,560, and 480 labor-hours per week, respectively. How many
of each style shirt must be produced each week for the plant to operate at full capacity?
Style A Style B Style C Time Available
Cutting department 0.2 hr 0.4 hr 0.3 hr 1,160 hr
Sewing department 0.3 hr 0.5 hr 0.4 hr 1,560 hr
Packaging department 0.1 hr 0.2 hr 0.1 hr 480 hr
SOLUTION
Let
Then
x Number of style A shirts produced per week
y Number of style B shirts produced per week
z Number of style C shirts produced per week
0.2x 0.4y 0.3z 1,160
0.3x 0.5y 0.4z 1,560
0.1x 0.2y 0.1z 480
Cutting department
Sewing department
Packaging department