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Chapter 4: Problem 10

The Fan Cost Index (FCI) represents the cost of four average-price tickets, four small soft drinks, two small beers, four hot dogs, parking for one car. two game programs, and two souvenir caps to a sporting event. (Source: Team Marketing Report.) (IMAGE CANNOT COPY) In \(2009,\) the FCI prices for Major League Baseball and the National Football League totaled \(\$ 609.53 .\) The football \(\mathrm{FCl}\) was \(\$ 215.75\) more than that of baseball. What were the FCIs for these sports?

Short Answer

Expert verified
The baseball FCI is \( 196.89 \) and the football FCI is \( 412.64 \).

Step by step solution

01

- Define Variables

Let the FCI for baseball be denoted as \( B \) and the FCI for football be denoted as \( F \).
02

- Set Up Equations

We have two pieces of information:1. The total FCI for both sports is \( B + F = 609.53 \).2. The football FCI is \( 215.75 \) more than the baseball FCI: \( F = B + 215.75 \).
03

- Substitute and Solve

Substitute \( F = B + 215.75 \) into the first equation: \[ B + (B + 215.75) = 609.53 \] Combine like terms: \[ 2B + 215.75 = 609.53 \] Solve for \( B \): \[ 2B = 609.53 - 215.75 \] \[ 2B = 393.78 \] \[ B = 196.89 \].
04

- Find Football FCI

Now use the value of \( B \) to find \( F \): \[ F = B + 215.75 \] \[ F = 196.89 + 215.75 \] \[ F = 412.64 \].
05

- Verify the Solution

Check that the values of \( B \) and \( F \) add up to the total FCI: \[ 196.89 + 412.64 = 609.53 \] This confirms our solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Linear Equations
A system of linear equations is a set of two or more linear equations with the same variables.
In this exercise, the variables used are the Fan Cost Index (FCI) for baseball denoted as B and for football denoted as F.
Solving a system of equations means finding the values of the variables that satisfy all equations simultaneously.
For this exercise, the two equations given are:
1. The total FCI for both sports is \(B + F = 609.53\).
2. The football FCI is \(215.75\) more than the baseball FCI: \(F = B + 215.75\).
These equations represent the relationship between the FCI prices of baseball and football.
Substitution Method
The substitution method is a technique used to solve a system of linear equations.
This involves solving one of the equations for one variable in terms of the other variable and then substituting this expression into the other equation.
In this exercise, we solve for F in terms of B from the second equation: \(F = B + 215.75\).
Next, we substitute this expression into the first equation: \(B + (B + 215.75) = 609.53\).
This results in a single equation with one variable, which can then be solved using regular arithmetic operations.
This method simplifies the solution process by reducing the number of equations.
Arithmetic Operations
Arithmetic operations are the basic calculations we perform in math such as addition, subtraction, multiplication, and division.
These operations are essential in solving linear equations.
In this exercise, addition and subtraction are primarily used to combine like terms and isolate variables.
For example, after substituting F in the equation, we get \(B + (B + 215.75) = 609.53\).
Combining like terms, we perform addition: \(2B + 215.75 = 609.53\).
Next, we subtract \(215.75\) from both sides: \(2B = 393.78\).
Finally, we divide by 2 to solve for B: \(B = 196.89\).
Similarly, arithmetic operations help us find F using B: \(F = 196.89 + 215.75\).

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