To set up a system of equations for the problem, we can define:
- Let \( p \) be the number of apples needed for one apple pie.
- Let \( t \) be the number of apples needed for one apple tart.
Now we can create equations based on the information given:
For Jada:
- She baked 8 apple pies and 1 apple tart, which used a total of 70 apples. This gives us our first equation: \[ 8p + 1t = 70 \tag{1} \]
For Laura:
- She made 3 apple pies and 5 apple tarts, which used a total of 54 apples. This gives us our second equation: \[ 3p + 5t = 54 \tag{2} \]
Now we have the following system of equations:
- \( 8p + t = 70 \)
- \( 3p + 5t = 54 \)
Next, we can solve this system using substitution or elimination. Let's use substitution with the first equation. From equation (1), we can express \( t \) in terms of \( p \): \[ t = 70 - 8p \tag{3} \]
Now, substitute equation (3) into equation (2): \[ 3p + 5(70 - 8p) = 54 \]
Distribute \( 5 \) in the equation: \[ 3p + 350 - 40p = 54 \]
Combine like terms: \[ -37p + 350 = 54 \]
Subtract 350 from both sides: \[ -37p = 54 - 350 \] \[ -37p = -296 \]
Now, divide by -37: \[ p = \frac{296}{37} = 8 \]
So, each apple pie requires \( 8 \) apples.
Next, we substitute \( p \) back into equation (3) to find \( t \): \[ t = 70 - 8(8) \] \[ t = 70 - 64 \] \[ t = 6 \]
Thus, each apple tart requires \( 6 \) apples.
Summarizing our results, we have:
An apple pie uses 8 apples and an apple tart requires 6 apples.
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