Let [tex] a [/tex] be the price of an adult ticket, and [tex] c [/tex] the price of a child ticket.
The sentence "Three adults and four children must pay $122 for tickets" translates to [tex] 3a+4c = 122 [/tex]
The sentence "Two adults and three children must pay $87" translates to [tex] 2a+3c = 87 [/tex]
Which leads to the linear system
[tex] \begin{cases} 3a+4c = 122\\2a+3c = 87 \end{cases} [/tex]
You can solve this system as you prefer, for example you can multiply the first equation by 2 and the second by 3 to get
[tex] \begin{cases} 6a+8c = 244\\6a+9c = 261\end{cases} [/tex]
Now subtract the first from the second:
[tex] (6a+9c) - (6a+8c) = 261-244 \iff c = 17 [/tex]
Now plug this value for c in any of the equations, for example the first:
[tex] 3a+4c = 122 \iff 3a+4\cdot 17 = 122 \iff 3a+ 68=122 \iff 3a = 54 \iff a = 18 [/tex]
