The entry fee for a carnival is different for adults and seniors. Children get in for free. Suppose that on the first day, the carnival admits 25 adults, 10 seniors, and 34 children. On the second day, the carnival admits 33 adults, 14 seniors, and 55 children. If the carnival collected $450 on the first day and $602 on the second day, what was the price, in dollars, of an adult entrance fee?

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Since children get in for free, we can ignore them when considering total entrance fee.

$$ \small 25 \text{ adults} \cdot \text{cost of adult ticket} + 10 \text{ seniors}\cdot \text{cost of senior ticket} = \text{total costs} $$ $$ \small \colorbox{aqua}{25A + 10S = 450} $$ $$ \small33 \text{ adults} \cdot \text{cost of adult ticket} + 14 \text{ seniors}\cdot \text{cost of senior ticket} = \text{total costs} $$ $$\small \colorbox{yellow}{ 33A + 14S = 602} $$

We can use substitution or elimination to solve. Here, we show substitution. We want to find the number of adult tickets, so let's solve for \(S\) and substitute it into the other equation.

$$ 25A+10S=450$$ $$ 10S=-25A+450 $$ $$ 2S=-5A+90 $$ $$ \colorbox{aqua}{14S = -35A + 630}$$

Note we could have solved for \(S\) rather than \(14S\), but then we would have to work with fractions.

$$\colorbox{yellow}{ 33A + 14S = 602} $$ $$\colorbox{yellow}{ 33A + \colorbox{aqua}{-35A + 630} = 602} $$ $$ -2A+630=602$$ $$ -2A=-28 $$ $$ A = \boxed{14} $$

It would take some time to test the answer choices, but it is still a viable option. Substitute the given options into the price of adult tickets and check if it satisfies the question's conditions.