To solve the problem of determining the number of children's and adult's tickets sold at the amusement park, we can use a system of linear equations. Here are the steps to set it up and solve it:
1. Define the variables:
- Let [tex]\(x\)[/tex] be the number of children's tickets sold.
- Let [tex]\(y\)[/tex] be the number of adult's tickets sold.
2. Form the system of linear equations:
- The total number of attendees is 25,000. This gives us the first equation:
[tex]\[
x + y = 25{,}000
\][/tex]
- The total revenue earned is [tex]$600,000. Since children's tickets cost $[/tex]15 and adult's tickets cost $35, we can write the second equation:
[tex]\[
15x + 35y = 600{,}000
\][/tex]
3. Set up the coefficient matrix, variable matrix, and solution matrix:
- Coefficient matrix [tex]\(A\)[/tex]:
[tex]\[
A = \begin{pmatrix} 1 & 1 \\ 15 & 35 \end{pmatrix}
\][/tex]
- Variable matrix [tex]\(X\)[/tex]:
[tex]\[
X = \begin{pmatrix} x \\ y \end{pmatrix}
\][/tex]
- Solution matrix [tex]\(B\)[/tex]:
[tex]\[
B = \begin{pmatrix} 25{,}000 \\ 600{,}000 \end{pmatrix}
\][/tex]
Therefore, we have:
[tex]\[
AX = B
\][/tex]
4. Solve the system of equations:
By solving the system of linear equations, we find that:
[tex]\[
x = 13{,}750 \quad \text{and} \quad y = 11{,}250
\][/tex]
5. Interpret the solution:
- The number of children's tickets sold is 13,750.
- The number of adult's tickets sold is 11,250.
Thus, we have determined that on that particular day, the amusement park sold 13,750 children's tickets and 11,250 adult's tickets.
