Answer :
Certainly! Let's solve the problem step-by-step.
We need to find the price of one senior citizen ticket and one student ticket. We have some information about the ticket sales over two days, which we can translate into a system of equations.
1. Define Variables:
- Let [tex]\( x \)[/tex] be the price of a senior citizen ticket.
- Let [tex]\( y \)[/tex] be the price of a student ticket.
2. Set Up Equations:
- First Day: The school sold 1 senior citizen ticket and 7 student tickets for a total of [tex]$116.
- This gives us the equation: \( x + 7y = 116 \).
- Second Day: The school sold 4 senior citizen tickets and 1 student ticket for a total of $[/tex]26.
- This gives us the equation: [tex]\( 4x + y = 26 \)[/tex].
3. Solve the System of Equations:
We have the following system:
[tex]\[
\begin{align*}
x + 7y &= 116 \\
4x + y &= 26
\end{align*}
\][/tex]
We can solve this system by using methods such as substitution or elimination. In this case, I'll describe the general steps using elimination:
- Multiply the first equation by 4 to align the coefficients of [tex]\( x \)[/tex]:
[tex]\[
4(x + 7y) = 4 \times 116 \Rightarrow 4x + 28y = 464
\][/tex]
- Now subtract the second equation:
[tex]\[
(4x + 28y) - (4x + y) = 464 - 26 \\
27y = 438
\][/tex]
- Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{438}{27} = \frac{146}{9}
\][/tex]
- Substitute [tex]\( y = \frac{146}{9} \)[/tex] back into one of the original equations to find [tex]\( x \)[/tex]:
[tex]\[
x + 7 \left( \frac{146}{9} \right) = 116
\][/tex]
[tex]\[
x + \frac{1022}{9} = 116
\][/tex]
[tex]\[
x = 116 - \frac{1022}{9}
\][/tex]
[tex]\[
x = \frac{1044}{9} - \frac{1022}{9} = \frac{22}{9}
\][/tex]
Thus, the price of one senior citizen ticket is [tex]\(\frac{22}{9}\)[/tex] dollars, and the price of one student ticket is [tex]\(\frac{146}{9}\)[/tex] dollars.
We need to find the price of one senior citizen ticket and one student ticket. We have some information about the ticket sales over two days, which we can translate into a system of equations.
1. Define Variables:
- Let [tex]\( x \)[/tex] be the price of a senior citizen ticket.
- Let [tex]\( y \)[/tex] be the price of a student ticket.
2. Set Up Equations:
- First Day: The school sold 1 senior citizen ticket and 7 student tickets for a total of [tex]$116.
- This gives us the equation: \( x + 7y = 116 \).
- Second Day: The school sold 4 senior citizen tickets and 1 student ticket for a total of $[/tex]26.
- This gives us the equation: [tex]\( 4x + y = 26 \)[/tex].
3. Solve the System of Equations:
We have the following system:
[tex]\[
\begin{align*}
x + 7y &= 116 \\
4x + y &= 26
\end{align*}
\][/tex]
We can solve this system by using methods such as substitution or elimination. In this case, I'll describe the general steps using elimination:
- Multiply the first equation by 4 to align the coefficients of [tex]\( x \)[/tex]:
[tex]\[
4(x + 7y) = 4 \times 116 \Rightarrow 4x + 28y = 464
\][/tex]
- Now subtract the second equation:
[tex]\[
(4x + 28y) - (4x + y) = 464 - 26 \\
27y = 438
\][/tex]
- Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{438}{27} = \frac{146}{9}
\][/tex]
- Substitute [tex]\( y = \frac{146}{9} \)[/tex] back into one of the original equations to find [tex]\( x \)[/tex]:
[tex]\[
x + 7 \left( \frac{146}{9} \right) = 116
\][/tex]
[tex]\[
x + \frac{1022}{9} = 116
\][/tex]
[tex]\[
x = 116 - \frac{1022}{9}
\][/tex]
[tex]\[
x = \frac{1044}{9} - \frac{1022}{9} = \frac{22}{9}
\][/tex]
Thus, the price of one senior citizen ticket is [tex]\(\frac{22}{9}\)[/tex] dollars, and the price of one student ticket is [tex]\(\frac{146}{9}\)[/tex] dollars.
