Solving a System of Linear Equations: Aircraft Seating Capacities

Q1. One aircraft has a capacity of 55 more seats than a second aircraft. A third aircraft has 65 seats less than twice the seating capacity of the second aircraft. If their total number of seats is 1850, find the number of seats for each aircraft.

Background

Topic: Systems of Linear Equations

This question tests your ability to translate a real-world scenario into algebraic expressions and solve a system of equations. You are asked to find the seating capacities of three aircrafts based on relationships and a total sum.

Key Terms and Formulas

• Let $x$, $y$, and $z$ represent the capacities of the first, second, and third aircrafts, respectively.

• System of equations: A set of equations with multiple variables that can be solved simultaneously.

• Key expressions:

  • $x = y + 55$ (first aircraft has 55 more seats than the second)

  • $z = 2y - 65$ (third aircraft has 65 less than twice the second)

  • $x + y + z = 1850$ (total seats)

Step-by-Step Guidance

1. Express all variables in terms of $y$ (the second aircraft): $x = y + 55$ $z = 2y - 65$

2. Write the total seats equation using these expressions: $(y + 55) + y + (2y - 65) = 1850$

3. Combine like terms to simplify the equation: $y + 55 + y + 2y - 65 = 1850$ $4y + 55 - 65 = 1850$

4. Move constants to the right side and solve for $y$: $4y + 55 - 65 = 1850$ $4y - 10 = 1850$ $4y = 1860$

Try solving on your own before revealing the answer!

Final Answer:

$y = 465$

$x = y + 55 = 520$

$z = 2y - 65 = 865$

The first aircraft has 520 seats, the second aircraft has 465 seats, and the third aircraft has 865 seats. This was found by substituting the value of $y$ into the expressions for $x$ and $z$.