Question

ann marie jones has been pricing train fares for a group trip to new york. three adults and four children must pay $129. two adults and three children must pay $92. find the price of the adult’s ticket and the price of a child’s ticket.
the price of a child’s ticket is $\square$.
the price of an adult’s ticket is $\square$.

Explanation:

Step1: Define Variables

Let \( x \) be the price of an adult's ticket (in dollars) and \( y \) be the price of a child's ticket (in dollars).

Step2: Set Up Equations

From the problem, we have two equations:

1. For three adults and four children: \( 3x + 4y = 129 \)
2. For two adults and three children: \( 2x + 3y = 92 \)

Step3: Solve the System of Equations

We can use the elimination method. First, multiply the first equation by 2 and the second equation by 3 to make the coefficients of \( x \) equal:

• \( 2(3x + 4y) = 2(129) \) gives \( 6x + 8y = 258 \)
• \( 3(2x + 3y) = 3(92) \) gives \( 6x + 9y = 276 \)

Now, subtract the first new equation from the second new equation:
\( (6x + 9y) - (6x + 8y) = 276 - 258 \)
\( 6x + 9y - 6x - 8y = 18 \)
\( y = 18 \)

Step4: Find \( x \)

Substitute \( y = 18 \) into the second original equation \( 2x + 3y = 92 \):
\( 2x + 3(18) = 92 \)
\( 2x + 54 = 92 \)
Subtract 54 from both sides: \( 2x = 92 - 54 = 38 \)
Divide by 2: \( x = \frac{38}{2} = 19 \)

Answer:

The price of a child’s ticket is $18.
The price of an adult’s ticket is $19.