To derive the equation for the total charge \( y \) for a delivery of \( x \) cookies, we first note that there is a fixed delivery fee of $5, and the remaining charge is based on the number of cookies delivered.
From the information provided, we can define the total charge equation as follows:
\[ y = mx + b \]
where:
- \( y \) is the total charge,
- \( x \) is the number of cookies delivered,
- \( m \) is the cost per cookie,
- \( b \) is the fixed delivery fee ($5).
Given that the delivery fee is $5, we can express that as:
\[ b = 5 \]
Now, we'll calculate the cost per cookie (\( m \)). We have two scenarios based on the deliveries:
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For 12 cookies, the total charge is $20: \[ 20 = 5 + 12m \] Subtracting 5 from both sides gives: \[ 15 = 12m \] Dividing by 12: \[ m = \frac{15}{12} = 1.25 \text{ (cost per cookie)} \]
-
For 18 cookies, the total charge is $27.50: \[ 27.50 = 5 + 18m \] Subtracting 5 from both sides gives: \[ 22.50 = 18m \] Dividing by 18: \[ m = \frac{22.50}{18} = 1.25 \text{ (cost per cookie)} \]
Both calculations for \( m \) confirm that the cost per cookie is indeed $1.25.
Putting this all together, the equation representing the total charge \( y \) for \( x \) cookies is:
\[ y = 1.25x + 5 \]
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