Let's denote the price of a bag of popcorn as \( p \) and the price of a candy as \( c \).
From the information given:
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Lincoln's purchase gives us the equation: \[ 2p + 4c = 34 \]
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Zoey's purchase gives us the equation: \[ 6p + 5c = 74 \]
Now we have a system of two equations:
\[ \begin{align*} (1) & \quad 2p + 4c = 34 \ (2) & \quad 6p + 5c = 74 \end{align*} \]
We can simplify the first equation. Divide the entire equation by 2:
\[ p + 2c = 17 \quad \text{(3)} \]
Now we will express \( p \) in terms of \( c \) from equation (3):
\[ p = 17 - 2c \quad \text{(4)} \]
Next, we will substitute equation (4) into equation (2):
\[ 6(17 - 2c) + 5c = 74 \]
Distributing the 6 gives:
\[ 102 - 12c + 5c = 74 \]
Combining like terms results in:
\[ 102 - 7c = 74 \]
Now, we will isolate \( c \):
\[ -7c = 74 - 102 \] \[ -7c = -28 \]
Dividing both sides by -7:
\[ c = 4 \]
Now that we have the value of \( c \), we can substitute it back into equation (4) to find \( p \):
\[ p = 17 - 2(4) \] \[ p = 17 - 8 \] \[ p = 9 \]
The solutions for the prices are:
\[ \text{Price of a bag of popcorn } (p) = 9 \] \[ \text{Price of a candy } (c) = 4 \]
Finally, we conclude that the price of each bag of popcorn is $9 and the price of each candy is $4.