Chapter 3 Pair of Linear Equations in Two Variables

Case Study 1: Transportation and Cost Analysis

Case Description:
A transportation company provides services between two cities, A and B. It operates two types of vehicles: buses and mini-vans. Each bus has a capacity of 50 passengers and charges $200 per trip, while each mini-van has a capacity of 15 passengers and charges $100 per trip. Due to fluctuating demand, the company needs to determine the optimal mix of buses and mini-vans to meet a daily target of 350 passengers while keeping the total number of trips at 10.

Let $x$ represent the number of bus trips and $y$ represent the number of mini-van trips. The company needs to set up a pair of linear equations based on the capacity and cost requirements to make informed decisions on how to allocate its resources effectively.

MCQs:

1. Which of the following equations represents the capacity constraint based on the number of passengers?

   • A) $50x + 15y = 350$
   • B) $15x + 50y = 350$
   • C) $15x + 50y = 10$
   • D) $50x + 100y = 350$

2. What equation represents the total trip constraint?

   • A) $x + y = 350$
   • B) $x - y = 10$
   • C) $x + y = 10$
   • D) $10x + y = 350$

3. If the company decides to operate 4 bus trips, how many mini-van trips are needed to meet the passenger target?

   • A) 5
   • B) 6
   • C) 4
   • D) 2

4. What would be the cost of service if the company operates 3 bus trips and 7 mini-van trips?

   • A) $1,600
   • B) $1,500
   • C) $1,700
   • D) $1,800

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Case Study 2: Business and Profit Maximization

Case Description:
A company manufactures two products, P and Q. For each unit of product P, the company earns a profit of $20, and for each unit of product Q, it earns a profit of $30. Due to resource constraints, the company can produce a total of 100 units per day. Additionally, due to market demand, the company needs to ensure that the number of units of product Q does not exceed twice the number of units of product P. To maximize profit, the company sets up a system of linear equations that will help determine the optimal number of units for each product.

Let $x$ be the number of units of product P and $y$ be the number of units of product Q.

MCQs:

1. Which of the following equations represents the production constraint?

   • A) $x + y = 100$
   • B) $x + y \leq 100$
   • C) $x - y = 100$
   • D) $2x + y = 100$

2. Which equation represents the market demand constraint for products P and Q?

   • A) $y = 2x$
   • B) $y \geq 2x$
   • C) $y \leq 2x$
   • D) $y \geq x$

3. If the company produces 40 units of product P, what is the maximum number of units of product Q that can be produced under the market demand constraint?

   • A) 60 units
   • B) 80 units
   • C) 100 units
   • D) 50 units

4. How much profit does the company make if it produces 30 units of product P and 70 units of product Q?

   • A) $2,500
   • B) $2,400
   • C) $2,700
   • D) $2,600

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Case Study 3: Cost of Catering Services

Case Description:
A caterer provides two types of meal packages for an event: standard and deluxe. The standard package costs $40 per meal, and the deluxe package costs $70 per meal. A client orders a total of 100 meals, and the caterer charges a total of $5,000 for the order. The caterer must now determine the number of standard and deluxe packages included in this order to assess future pricing strategies.

Let $x$ represent the number of standard packages and $y$ represent the number of deluxe packages ordered.

MCQs:

1. What is the equation representing the total number of meals ordered?

   • A) $x + y = 5,000$
   • B) $40x + 70y = 5,000$
   • C) $x + y = 100$
   • D) $70x + 40y = 5,000$

2. Which equation represents the total cost of the meals?

   • A) $40x + 70y = 5,000$
   • B) $x + y = 5,000$
   • C) $70x + 40y = 100$
   • D) $x - y = 5,000$

3. If the client ordered 60 standard packages, how many deluxe packages were ordered?

   • A) 20
   • B) 30
   • C) 40
   • D) 50

4. If the client had ordered 40 deluxe packages, what would have been the cost of the order?

   • A) $3,200
   • B) $2,800
   • C) $5,000
   • D) $4,600

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Case Study 4: Agricultural Land Use Planning

Case Description:
A farmer has a plot of land on which they want to grow wheat and corn. The land is divided so that each acre of wheat yields $100 in profit, while each acre of corn yields $150 in profit. The farmer can use up to 20 acres of land, with the constraint that the number of acres of corn must be at least double the number of acres of wheat to meet market demands. The farmer needs to set up equations to determine the maximum profit achievable based on these conditions.

Let $x$ represent the acres of wheat, and $y$ represent the acres of corn.

MCQs:

1. Which of the following equations represents the land constraint?

   • A) $x + y = 20$
   • B) $x + y \leq 20$
   • C) $x + y \geq 20$
   • D) $x - y = 20$

2. Which inequality represents the constraint for corn to meet market demand?

   • A) $y \leq 2x$
   • B) $y \geq 2x$
   • C) $y = 2x$
   • D) $y = x$

3. If the farmer grows 5 acres of wheat, how many acres of corn are needed to satisfy market demand?

   • A) 10 acres
   • B) 12 acres
   • C) 15 acres
   • D) 8 acres

4. What is the maximum profit if the farmer grows 6 acres of wheat and 12 acres of corn?

   • A) $2,400
   • B) $2,100
   • C) $3,000
   • D) $2,700

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Case Study 5: Movie Theater Seating Allocation

Case Description:
A movie theater has two sections, A and B. Section A seats cost $150 each, and Section B seats cost $100 each. During a show, a total of 500 seats are sold, generating $60,000 in revenue. The theater management needs to find out how many seats from each section were sold to adjust future pricing and optimize seat allocation.

Let $x$ represent the number of seats sold in Section A, and $y$ represent the number of seats sold in Section B.

MCQs:

1. What equation represents the total number of seats sold?

   • A) $x + y = 500$
   • B) $x - y = 500$
   • C) $150x + 100y = 500$
   • D) $x + y = 60,000$

2. What equation represents the total revenue generated?

   • A) $x + y = 60,000$
   • B) $150x + 100y = 500$
   • C) $150x + 100y = 60,000$
   • D) $x - y = 60,000$

3. If 200 seats were sold in Section A, how many seats were sold in Section B?

   • A) 300 seats
   • B) 200 seats
   • C) 100 seats
   • D) 150 seats

4. What was the revenue generated if 250 seats were sold in Section B?

   • A) $55,000
   • B) $50,000
   • C) $60,000
   • D) $45,000