Problem 43 - Beginning and Intermediate Algebra

Question

The three NBA teams with the highest revenues in \(2002-2003\) were the New York Knicks, the Los Angeles Lakers, and the Chicago Bulls. Their revenues totalled \(\$ 428\) million. The Lakers took in \(\$ 30\) million more than the Bulls, and the Knicks took in \(\$ 11\) million more than the Lakers. Determine the revenue of each team during the \(2002-2003\) season. (Source. Forbes Feb. \(16,2004, p .66\)

Step-by-Step Solution

Answer

In the 2002-2003 season, the Lakers had a revenue of \(129\) million, the Knicks had a revenue of \(140\) million, and the Bulls had a revenue of \(99\) million.

Step 1: Set up the Variables

Let's assign a variable for each team's revenue: x = revenue of the Lakers y = revenue of the Knicks z = revenue of the Bulls

Step 2: Form Equations from the Problem Statement

We are given: 1. Total revenue of the three teams: x + y + z = 428 2. Lakers took in \(30\) million more than the Bulls: x = z + 30 3. Knicks took in \(11\) million more than the Lakers: y = x + 11

Step 3: Write the System of Equations

The system of equations can be written as: 1. x + y + z = 428 2. x - z = 30 3. y - x = 11

Step 4: Solve the System of Equations

We will solve the system using the method of substitution. Firstly we can substitute the second equation into the first equation: (x + 30) + y + x = 428 This simplifies to: 2x + y + 30 = 428 Now let's substitute the third equation into the new simplified equation: 2x + (x + 11) + 30 = 428 This simplifies to: 3x + 41 = 428

Step 5: Solve for x (Lakers' Revenue)

To find the value of x, we can subtract 41 from both sides of the equation and then divide by 3: 3x = 387 x = 129 So, the Lakers took in \(129\) million in revenue during the 2002-2003 season.

Step 6: Solve for z (Bulls' Revenue)

Use the value of x in equation 2: x - z = 30 129 - z = 30 Subtract 129 from both sides: -z = -99 Divide by -1: z = 99 So, the Bulls took in \(99\) million in revenue during the 2002-2003 season.

Step 7: Solve for y (Knicks' Revenue)

Use the value of x in equation 3: y - x = 11 y - 129 = 11 Add 129 to both sides: y = 140 The Knicks took in \(140\) million in revenue during the 2002-2003 season.

Step 8: Final Answer

In conclusion, the revenues of the three NBA teams during the 2002-2003 season were: Lakers: \(129\) million Knicks: \(140\) million Bulls: \(99\) million

Key Concepts

Algebra

Algebra is the foundation of solving problems involving system of equations like the one we have here. It involves using symbols and letters to represent numbers in equations and expressions. This allows us to find unknown values by working logically through expressions and equations.

In this exercise about NBA team revenues, we use algebra to set up equations based on the information given about the teams' earnings. It is crucial to understand how to translate a word problem into algebraic equations. For example:

• "The Lakers took in \(30\) million more than the Bulls" translates mathematically to \( x = z + 30 \).
• "The Knicks took in \(11\) million more than the Lakers" becomes \( y = x + 11 \).

Algebra helps us structure the problem in a way that makes it easier to solve step by step.

Substitution Method

The substitution method is a technique often used to solve systems of equations like the one in our problem about the NBA teams. It involves solving an equation for one variable and then substituting that expression into another equation. This eliminates one variable, simplifying the system step by step.

To see substitution in action:

• First, we express one variable in terms of the others, like \( x = z + 30 \), and substitute it into another equation.
• By substituting \( x = z + 30 \) into \( x + y + z = 428 \), we get \( (z + 30) + y + z = 428 \).
• The goal is to have one equation in one variable, which is easier to solve.

The substitution method simplifies complex systems by breaking them down into single-variable equations that can be solved sequentially.

Revenue Calculation

Revenue calculation in algebra problems involves determining the amount of money generated, which is crucial in various real-world applications, including sports teams' financial evaluations. To handle calculations effectively, it's important to set up a clear system of equations based on given information.

In the NBA revenue problem:

• We have total revenue and relationships between individual team revenues, translating into three key equations.
• We then use the substitution method to isolate and solve for a variable, like determining the revenue for the Lakers first, then the Bulls, and finally the Knicks.

Ultimately, solving each equation step by step provides the exact revenue each NBA team garnered during the specified season, integrating mathematical calculations with logical reasoning.

NBA Team Revenues

Understanding NBA team revenues isn't just about numbers; it's about interpreting financial data and making it meaningful. Teams' revenues reflect their market size, fan base, and overall success, shedding light on their operations and financial health.

In the context of the 2002-2003 season, calculating the revenues for the Knicks, Lakers, and Bulls reveals more than just figures.

• The Lakers' revenue, influenced by their popularity and performance, was \(129\) million.
• The Knicks, despite performance variances, netted \(140\) million, showcasing their robust market presence.
• The Bulls, in a rebuilding phase post-Michael Jordan, accrued \(99\) million, demonstrating the shifting dynamics of team success and revenue.

Analyzing these revenues lets us understand economic factors affecting sports teams and appreciate the algebraic methods we use to derive such insights.