Let [tex]\( a \)[/tex] be the price of an adult ticket and [tex]\( c \)[/tex] be the price of a child ticket.
We are given two equations based on the information provided:
1) [tex]\( 2a + 4c = 38.00 \)[/tex]
2) [tex]\( 3a + 3c = 40.50 \)[/tex]
First, we can simplify these equations to make them easier to solve. Divide the first equation by 2:
[tex]\[ a + 2c = 19 \][/tex]
Now, divide the second equation by 3:
[tex]\[ a + c = 13.50 \][/tex]
We now have a system of linear equations:
[tex]\[
\begin{cases}
a + 2c = 19 \\
a + c = 13.50
\end{cases}
\][/tex]
To solve for [tex]\(a\)[/tex] and [tex]\(c\)[/tex], we can eliminate [tex]\(a\)[/tex] by subtracting the second equation from the first:
[tex]\[
(a + 2c) - (a + c) = 19 - 13.50
\][/tex]
Simplify this equation:
[tex]\[
a + 2c - a - c = 5.50 \\
c = 5.50
\][/tex]
Now that we have [tex]\( c \)[/tex], substitute [tex]\( c = 5.50 \)[/tex] back into the equation [tex]\( a + c = 13.50 \)[/tex]:
[tex]\[
a + 5.50 = 13.50
\][/tex]
Solve for [tex]\(a\)[/tex]:
[tex]\[
a = 13.50 - 5.50 \\
a = 8.00
\][/tex]
Thus, the values of [tex]\( a \)[/tex] and [tex]\( c \)[/tex] are:
[tex]\[
a = 8.00 \quad \text{and} \quad c = 5.50
\][/tex]
So, an adult ticket costs [tex]\( \mathbf{8.00} \)[/tex] and a child ticket costs [tex]\( \mathbf{5.50} \)[/tex].
