To find the prices for adults and children at the movie theater, we can set up a system of equations based on the information provided.
Let's denote:
- \( a \) as the price of an adult ticket
- \( c \) as the price of a child ticket
From the information given, we can create the following equations:
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From Friday's ticket sales: \[ 10a + 25c = 670 \tag{1} \]
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From Sunday's ticket sales: \[ 5a + 10c = 290 \tag{2} \]
We can simplify Equation (2) by dividing each term by 5: \[ a + 2c = 58 \tag{3} \]
Now we have a simpler equation to work with. Next, we can solve Equation (3) for \( a \): \[ a = 58 - 2c \tag{4} \]
Now, we will substitute Equation (4) into Equation (1): \[ 10(58 - 2c) + 25c = 670 \] Distributing 10 gives us: \[ 580 - 20c + 25c = 670 \] Combining like terms: \[ 580 + 5c = 670 \] Subtracting 580 from both sides: \[ 5c = 90 \] Dividing by 5: \[ c = 18 \]
Now that we have \( c \) (the cost of a child ticket), we can substitute it back into Equation (4) to find \( a \): \[ a = 58 - 2(18) \] Calculating this gives: \[ a = 58 - 36 = 22 \]
Thus, the prices for the tickets are:
- Adult ticket price \( a = 22 \) dollars
- Child ticket price \( c = 18 \) dollars
Final Answer:
- Adult ticket price: $22
- Child ticket price: $18
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