Problem 27 - College Algebra

Question

Two of the most expensive movies ever made were Titanic and Waterworld. The cost to make Titanic exceeded the cost to make Waterworld by \(\$ 40\) million. The combined cost to make the two movies was \(\$ 360\) million. Find the cost of making each of these movies.

Step-by-Step Solution

Answer

The cost of making Waterworld was \$160 million and the cost of making Titanic was \$200 million.

Step 1: Formulate the Equations

From the problem, it's stated that: The cost to make Titanic exceeded the cost to make Waterworld by $40 million, which we can write as \(T = W + 40\). It's also stated that: The combined cost to make the two movies was $360 million, which we can write as \(T + W = 360\). This gives us the system of equations: \(T = W + 40\) and \(T + W = 360\).

Step 2: Solve The Equations

Substitute the 1st equation \(T = W + 40\) into the 2nd equation: \(T + W = 360\). This gives us \(W + 40 + W = 360\), or \(2W + 40 = 360\). Solve it for W: \(2W = 320\), so \(W = 160\). Substituting \(W = 160\) into the first equation \(T = W + 40\), we get \(T = 200\).

Step 3: Interpret The Results

The solution \(W = 160\) and \(T = 200\) means that Waterworld cost $160 million and Titanic cost $200 million to make.

Key Concepts

Understanding Linear Equations

When we talk about **linear equations**, we're dealing with mathematical expressions that describe a straight line when graphed. Think of them as simple equations with variables that only appear to the first power. Linear equations are an essential part of algebra and are foundational concepts for understanding more complex mathematics.

• Linear equations take the form of: \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants.
• In our example problem around movie costs, we used linear equations to model the relationships between the costs.

Understanding linear equations means recognizing how variables relate to one another under set constraints. In our problem, the constraints were the sum of costs and the difference in costs. These relationships are easily expressed through linear equations, helping us solve for unknown values efficiently.

Exploring the Substitution Method

The **substitution method** is a powerful tool for solving systems of equations. It's a technique where you solve one equation for one variable and then substitute that into another equation. By doing this, you simplify the system to one equation with one variable, making it easier to solve.

• First, isolate one variable in one of the equations.
• Next, substitute this expression into the other equation.
• This allows you to find the value of one variable, which can then be used to find the other.

In our problem, we started with two equations: \( T = W + 40 \) and \( T + W = 360 \). We used the first equation to express \( T \) in terms of \( W \), and then substituted it into the second equation. This straightforward method allowed us to find \( W \), and subsequently \( T \). The substitution method is especially helpful when dealing with real-world problems, like our movie budget example, where relationships are straightforward but involve more than one variable.

Effective Problem-Solving Strategies

Engaging in **problem-solving** involves a strategic approach and some practice. In problems involving systems of equations, like our example with movie budgets, it's crucial to follow a step-by-step approach.

• Identify the variables: Clearly label what each variable represents.
• Formulate the equations: Translate the problem's conditions into mathematical equations.
• Choose a solving method: Depending on the problem, choose from methods like substitution or elimination. We chose substitution for the movie problem.
• Solve and interpret: Solve the equations and interpret the results in the context of the problem. This tells you if the solution makes sense.

Remember, solving problems is like solving a puzzle. Breaking it down methodically helps uncover the solution piece by piece. With practice, you'll find these strategies become intuitive.

Mastering Algebraic Expressions

**Algebraic expressions** are combinations of variables, numbers, and operations that represent values and relationships. They're crucial in forming equations and inequalities. In our movie cost problem, understanding algebraic expressions was key to setting up our equations.

• An algebraic expression can include constants, variables, addition, subtraction, multiplication, and division.
• Understanding the expression \( T = W + 40 \), means realizing "\( T \) exceeds \( W \) by 40."
• Algebraic expressions can be manipulated using algebraic rules to find unknown values.

Every algebraic expression has its role in forming equations that model real-world scenarios. Once you grasp how to construct and deconstruct these expressions, you gain the ability to solve numerous mathematical challenges. This flexibility is useful whether you're budgeting movies or solving equations in a math class.