let

a = the price of adult ticket

y = the price of youth ticket

Brianna's family spent $134 on 2 adult tickets and 3 youth tickets at an amusement park:

2a + 3y = 134

Macs family spent $146 on 3 adult tickets and 2 youth tickets:

3a + 2y = 146

by solving the system of equation:

2a + 3y = 134

3a + 2y = 146

we find:

a = $34

y = $22

click here to see the step by step solution of the system of equations:

the price of the youth ticket is $22.