Problem 19 - Intermediate Algebra

Question

\(X\) -Files, Will \& Grace, and Seinfeld are three of the most popular television shows of all time. The total number of episodes of these three shows is \(575 .\) There are 21 more episodes of \(X\) -Files than Seinfeld , and the difference between the number of episodes of Will \& Grace and Seinfeld is 14 Find the number of episodes of each show.

Step-by-Step Solution

Answer

X-Files: 201 episodes, Will & Grace: 194 episodes, Seinfeld: 180 episodes.

Step 1: Define Variables

Let's define three variables to represent the number of episodes for each show: let \( x \) be the number of episodes for X-Files, \( w \) for Will & Grace, and \( s \) for Seinfeld.

Step 2: Set Up Equations

Using the information given, we can set up the following equations:1. \( x + w + s = 575 \) (the total number of episodes)2. \( x = s + 21 \) (21 more episodes of X-Files than Seinfeld)3. \( w = s + 14 \) (14 more episodes of Will & Grace than Seinfeld).

Step 3: Substitute and Simplify

Substitute \( x = s + 21 \) and \( w = s + 14 \) into the first equation:\( (s + 21) + (s + 14) + s = 575 \). Simplifying gives: \( 3s + 35 = 575 \).

Step 4: Solve for Seinfeld

Solve for \( s \) by subtracting 35 from both sides of the equation, resulting in:\( 3s = 540 \). Then, divide by 3 to find \( s \):\( s = 180 \).

Step 5: Determine Episodes for X-Files

Using \( s = 180 \) in the equation for X-Files, \( x = s + 21 \), substitute \( s \):\( x = 180 + 21 = 201 \).

Step 6: Determine Episodes for Will & Grace

Substitute \( s = 180 \) into the equation for Will & Grace, \( w = s + 14 \):\( w = 180 + 14 = 194 \).

Key Concepts

Variable Substitution

Variable substitution is a powerful method used to solve systems of equations. In this method, we replace variables in one equation with expressions from other equations. This helps simplify our problem step-by-step.
Imagine you're given a puzzle that requires solving for multiple unknowns. Instead of tackling all unknowns at once, you address one at a time.

• First, you determine expressions for some of your variables using the given conditions.
• In this example, we know that the number of episodes of X-Files, denoted by \( x \), is 21 more than Seinfeld, \( s \). Hence, \( x = s + 21 \).
• The same logic applies to Will & Grace (\( w = s + 14 \)), which is 14 more episodes than Seinfeld.

These variable replacements allow us to express everything in terms of one variable, simplifying the solving process.

Linear Equations

Linear equations are foundational in algebra and involve variables raised only to the first power. These equations represent straight lines when graphed.
For our exercise with TV shows, we've set up linear equations to express the relationships between episodes:

• \( x + w + s = 575 \)
• \( x = s + 21 \)
• \( w = s + 14 \)

Each equation is straightforward, consistent with the properties of linear equations, meaning they're easy to solve. By transforming our word problem into linear equations, we harness a powerful way to find our solutions

When working with linear equations, remember that each transformation made must preserve the equation's equality. This is crucial for maintaining balance between expressions on either side of the equation sign.

Solving Word Problems

Word problems can seem tricky at first because they require you to translate real-world scenarios into mathematical equations. Here's a breakdown to make the task seamless:

• Start by understanding the scenario described. Identify what is being asked.
• Define your variables clearly, as they represent the unknowns you need to find. In this assignment, variables \( x \), \( w \), and \( s \) stood for the episodes of X-Files, Will & Grace, and Seinfeld, respectively.
• Convert the situation into mathematical expressions or equations. Notice how the relationships among the episodes were translated into equations.

After setting up your equations, follow through systematically to solve for each variable using appropriate methods, such as substitution or elimination. Keep practicing these steps, and you'll find that solving word problems makes much more sense and becomes considerably easier.