Problem 11 - Intermediate Algebra

Question

Airplanes. Together, a Delta B747 and a Delta B777 seat 681 passengers. If the B777 seats 125 less people than the B747, how many passengers does each seat? (Source: deltaskymag.com)

Step-by-Step Solution

Answer

The B747 seats 403 passengers, and the B777 seats 278 passengers.

Step 1: Define Variables

Let \( x \) be the number of passengers the B747 seats. Therefore, the B777 seats \( x - 125 \) passengers, as it sits 125 less passengers than the B747.

Step 2: Formulate Equation

Based on the problem, the total number of passengers that both airplanes can seat together is 681. Hence, our equation is given by: \[ x + (x - 125) = 681 \]

Step 3: Simplify Equation

First, simplify the equation by combining like terms: \[ 2x - 125 = 681 \]

Step 4: Solve for x

Add 125 to both sides of the equation to isolate terms with \( x \) on one side:\[ 2x - 125 + 125 = 681 + 125 \]\[ 2x = 806 \]

Step 5: Find x

Divide both sides of the equation by 2 to solve for \( x \):\[ x = \frac{806}{2} \]\[ x = 403 \]Thus, the B747 seats 403 passengers.

Step 6: Calculate B777 Seats

Use the value of \( x \) to find out how many passengers the B777 seats by substituting \( x = 403 \) into the expression for the B777:\[ x - 125 = 403 - 125 \]\[ = 278 \]So, the B777 seats 278 passengers.

Key Concepts

Variables in Algebra

In algebra, variables serve as placeholders in equations, making it easier to represent and solve real-world problems. When faced with a situation like an algebra word problem, identifying the variables is a crucial step. Consider the problem of finding out how many passengers a Delta B747 and a B777 can seat. By choosing a variable, such as \( x \), to represent the number of passengers the B747 can seat, we open the door to formulating equations and finding solutions.
Variables bridge the problem's narrative with mathematical expressions, providing clarity and simplifying complex problems. Once a variable is defined, it translates the problem's conditions into relatable mathematical language, enabling further steps like equation solving.

Equation Solving

Equation solving is a key part of algebra that involves finding the value of the variables that satisfy an equation. In our problem, after defining the variables, the next step is to create an equation based on the relationships given in the problem. We know the B747 seats \( x \) passengers and the B777 seats \( x - 125 \) passengers. Together, they seat 681 passengers, leading to the equation:

• \( x + (x - 125) = 681 \)

To solve the equation, we combine like terms and simplify it to:

• \( 2x - 125 = 681 \)

Next, isolate \( x \) by adding 125 to both sides:
\( 2x = 806 \)
Finally, divide by 2 to find \( x \):
\( x = 403 \)
Thus, the B747 seats 403 passengers. Equation solving helps us take a narrative from the real world, model it with math, and deduce solid numerical answers.

System of Equations

Though the original problem involves a single equation, it stems from the concept of a system of equations where several equations work together. In many cases, systems of equations consist of multiple equations that need to be solved together because they share variables. Here, our simplified approach involved one equation, but it showcases the idea that problems often have multiple relationships that can be woven into systems.
Exploring systems of equations, we often deal with methods like substitution or elimination to find solutions. Different systems might require creative approaches to determine how variables interrelate. In this case, we used a simplified approach by defining relationships and setting up an equation based on those, ensuring everything ties back to the central idea of solving interconnected relationships in algebra environments.