Question

three adults and four children must pay $122 for tickets. Two adults and three children must pay $87. what are the price of each ticket.

Answer 1

Let $a$ be the price of an adult ticket, and $c$ the price of a child ticket.

The sentence "Three adults and four children must pay $122 for tickets" translates to $3a+4c = 122$

The sentence "Two adults and three children must pay $87" translates to $2a+3c = 87$

Which leads to the linear system

$\begin{cases} 3a+4c = 122\\2a+3c = 87 \end{cases}$

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

$\begin{cases} 6a+8c = 244\\6a+9c = 261\end{cases}$

Now subtract the first from the second:

$(6a+9c) - (6a+8c) = 261-244 \iff c = 17$

Now plug this value for c in any of the equations, for example the first:

$3a+4c = 122 \iff 3a+4\cdot 17 = 122 \iff 3a+ 68=122 \iff 3a = 54 \iff a = 18$