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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.
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.
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