Question

1. to attend the state fair, one group of 10 adults and 12 children paid $126 for their tickets. another group of 2 adults and 18 children paid $72 for their tickets. what is the price for each adult ticket?
2. at the high school football game, a student purchased 3 tacos and 7 pieces of sour hard candy for $6.60. a second student purchased 4 tacos and 8 pieces of sour hard candy for $8.40. how much does it cost for each item?

Explanation:

Problem 2

Step1: Define variables

Let $x$ = adult ticket price, $y$ = child ticket price.

Step2: Set up equations

$$10x + 12y = 126$$
$$2x + 18y = 72$$

Step3: Simplify second equation

Divide by 2: $x + 9y = 36$, so $x = 36 - 9y$

Step4: Substitute into first equation

$$10(36 - 9y) + 12y = 126$$
$$360 - 90y + 12y = 126$$
$$360 - 78y = 126$$

Step5: Solve for y

$$-78y = 126 - 360$$
$$-78y = -234$$
$$y = \frac{-234}{-78} = 3$$

Step6: Solve for x

$$x = 36 - 9(3) = 36 - 27 = 9$$

Step1: Define variables

Let $t$ = taco price, $c$ = candy price.

Step2: Set up equations

$$3t + 7c = 6.60$$
$$4t + 8c = 8.40$$

Step3: Simplify second equation

Divide by 4: $t + 2c = 2.10$, so $t = 2.10 - 2c$

Step4: Substitute into first equation

$$3(2.10 - 2c) + 7c = 6.60$$
$$6.30 - 6c + 7c = 6.60$$
$$6.30 + c = 6.60$$

Step5: Solve for c

$$c = 6.60 - 6.30 = 0.30$$

Step6: Solve for t

$$t = 2.10 - 2(0.30) = 2.10 - 0.60 = 1.50$$

Answer:

The price of each adult ticket is $\$9$

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Problem 4