Problem 12 - Intermediate Algebra

Question

Denzel. As of October 2010, Denzel Washington's three top domestic grossing films, American Gangster, Remember the Titans, and The Pelican Brief, had earned a total of \(\$ 346.7\) million. If American Gangster earned \(\$ 14.5\) million more than Remember the Titans, and if Remember the Titans earned \(\$ 14.9\) million more than The Pelican Brief, how much did each film earn as of that date?

Step-by-Step Solution

Answer

'The Pelican Brief' earned \(100.8\) million, 'Remember the Titans' earned \(115.7\) million, and 'American Gangster' earned \(130.2\) million.

Step 1: Define Variables

Let \( x \) represent the earnings of 'The Pelican Brief'. Then 'Remember the Titans' earned \( x + 14.9 \) million and 'American Gangster' earned \( x + 14.9 + 14.5 = x + 29.4 \) million.

Step 2: Write the Equation

The total earnings from the three films is \( 346.7 \) million. Therefore, the equation is:\[ x + (x + 14.9) + (x + 29.4) = 346.7 \]

Step 3: Simplify the Equation

Combine like terms:\[ 3x + 44.3 = 346.7 \]

Step 4: Solve for x

Subtract 44.3 from both sides:\[ 3x = 302.4 \]Divide both sides by 3 to find \( x \):\[ x = 100.8 \]

Step 5: Find Each Film's Earnings

- 'The Pelican Brief' earned \( x \), which is \( 100.8 \) million.- 'Remember the Titans' earned \( x + 14.9 = 115.7 \) million.- 'American Gangster' earned \( x + 29.4 = 130.2 \) million.

Key Concepts

System of Equations

In algebra, a system of equations is a set of two or more equations that have common variables. The goal is to find a solution that satisfies all of the equations in the system. In this particular problem, we deal with only one equation involving multiple conditions. Nevertheless, the approach mimics solving a system because we have relationships between the variables that must satisfy the total sum. By carefully defining and working with these relationships, we can solve for the unknowns systematically.

Consider how each film's earnings are interlinked. This interconnected information forms the basis of our equation. Solving it will involve using all the given relationships, just as you would in a broader system of equations.

Variable Definition

Defining variables is the crucial first step in solving word problems like this one. We simplify the real-world scenario into mathematical terms by assigning symbols to represent unknown quantities. In our example, we start by letting \( x \) represent the earnings of 'The Pelican Brief'. By using this variable, we can express the earnings for the other films in relation to \( x \).

Next, we express the additional earnings of 'Remember the Titans' over 'The Pelican Brief' as \( x + 14.9 \). Similarly, 'American Gangster' is expressed as \( x + 29.4 \) because it earned \( 14.5 \) million more than 'Remember the Titans', making it \( 14.9 + 14.5 \) million more than 'The Pelican Brief'. By defining these variables, we create a clear path to formulating our equation.

Equation Solving

Once we have our variables, we use them to write an equation that reflects the total earnings from all three films equaling \( 346.7 \) million. The equation, \( x + (x + 14.9) + (x + 29.4) = 346.7 \), incorporates the information from the problem. Our task is to solve this equation for \( x \).

By simplifying the left-hand side to combine like terms, we rewrite the equation as \( 3x + 44.3 = 346.7 \). Then, we solve for \( x \) by first subtracting \( 44.3 \) from both sides to get \( 3x = 302.4 \), and finally divide by 3 to find \( x = 100.8 \). Solving the equation meticulously allows us to determine the earnings of each film accurately.

Linear Equations

Linear equations are equations in which the highest power of the variable is one. They form straight lines when graphed. In this problem, our equation \( 3x + 44.3 = 346.7 \) is an example of a linear equation.

Solving linear equations involves basic techniques:

• Combining like terms.
• Isolating variables.
• Performing arithmetic operations such as addition, subtraction, multiplication, and division.

By methodically applying these techniques, we solve for \( x \), making linear equations an essential tool for finding solutions in algebra word problems like this one.