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Difficulty: MediumSystems of Linear Equations

A theater sells adult tickets for xx dollars each and child tickets for yy dollars each. For a morning show, the theater sold 2 adult tickets and 1 child ticket, collecting a total of 1414 dollars. For an afternoon show, the theater sold 3 adult tickets and 2 child tickets, collecting a total of 2323 dollars. What is the cost, in dollars, of 1 adult ticket and 1 child ticket combined?

  1. A
    5
  2. B
    6
  3. 9Answer
  4. D
    19

Answer

The combined cost of 1 adult ticket and 1 child ticket is 9 dollars.
The correct answer is 9. Subtracting the equation representing the morning show (2x+y=142x + y = 14) from the equation representing the afternoon show (3x+2y=233x + 2y = 23) directly yields x+y=9x + y = 9. Alternatively, solving the system of equations by substitution or elimination gives x=5x = 5 (cost of an adult ticket) and y=4y = 4 (cost of a child ticket), and adding these two values results in 99.

Step-by-Step Solution

1
Write the system of linear equations based on the problem description.
The system of equations is:
2x+y=143x+2y=23\begin{aligned} 2x + y &= 14 \\ 3x + 2y &= 23 \end{aligned}
To mathematically represent the ticket sales and total revenue for the morning and afternoon shows.
2
Subtract the first equation from the second equation to find the value of x+yx + y directly.
(3x+2y)−(2x+y)=23−14  ⟹  x+y=9(3x + 2y) - (2x + y) = 23 - 14 \implies x + y = 9
Subtracting the equations isolates the expression x+yx + y immediately, which represents the combined cost of 1 adult ticket and 1 child ticket.

Key Concept

Solving systems of linear equations by elimination or subtraction to find a linear combination of variables.

Alternative Method

Solve the system using elimination: multiply the first equation by 2 to get 4x+2y=284x + 2y = 28. Subtract the second equation (3x+2y=233x + 2y = 23) from this new equation to find x=5x = 5. Substitute x=5x = 5 into the first equation to find y=4y = 4. Finally, add the values to get x+y=9x + y = 9.
Estimated Time:1m 30s
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Topics

Systems of Linear Equations

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