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Kayla's school is selling tickets to the annual dance competition. On the first day of ticket sales, the school sold 3 senior citizen tickets and 5 child tickets for a total of $70. The school took in $216 on the second day by selling 12 senior citizen tickets and 12 child tickets.



Let $x$ represent the price of a senior citizen ticket and $y$ represent the price of a child ticket.



Select the two equations that model this scenario.


  • $12x + 12y = 216$
  • $3x + 5y = 216$
  • $3x + 5y = 70$
  • $12x + 12y = 70$
  • $x + y = 70$

Verified Answer :

  • Define variables \(x\) for the price of a senior ticket and \(y\) for the price of a child ticket.
  • Derive the first equation from Day 1 sales: \(3x + 5y = 70\).
  • Derive the second equation from Day 2 sales: \(12x + 12y = 216\).
  • Select the matching options from the given list: \(\boxed{3x + 5y = 70, 12x + 12y = 216}\).

Explanation

  1. Analysis of the Problem

The problem describes the sale of two types of tickets, senior citizen tickets (at price \(x\)) and child tickets (at price \(y\)), over two days. To model this, we translate the verbal descriptions of the sales into algebraic equations. We are given two distinct scenarios, one for each day. We need to identify the equations that correspond to these scenarios.

  1. Formulating the Equation for Day 1

On the first day, the school sold 3 senior citizen tickets and 5 child tickets for a total revenue of \(\$70\). Since the price of a senior citizen ticket is \(x\) and the price of a child ticket is \(y\), the revenue from senior citizen tickets is \(3x\) and the revenue from child tickets is \(5y\). The sum of these amounts equals the total revenue:
\[3x + 5y = 70\]

  1. Formulating the Equation for Day 2

On the second day, the school sold 12 senior citizen tickets and 12 child tickets for a total revenue of \(\$216\). Similarly, the revenue from senior citizen tickets is \(12x\) and the revenue from child tickets is \(12y\). The total revenue equation is:
\[12x + 12y = 216\]

  1. Selecting the Correct Equations

Comparing our derived equations to the provided options:

  1. \(12x + 12y = 216\) (Matches Day 2)
  2. \(3x + 5y = 70\) (Matches Day 1)

These are the two equations that correctly model the ticket sales scenario.

Examples

Understanding how to build systems of equations is fundamental in managing finances. For example, if a store owner needs to track the revenue from two different products to ensure their pricing strategy is profitable, they can set up similar equations. By tracking the number of items sold and their respective prices, one can solve for unknown values and make data-driven decisions for future sales, just as in this ticket sale scenario.

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