Let's denote the flat fee for processing as \(b\) and the cost per pass as \(m\). The linear equation for the total cost can be represented as:
\[ y = mx + b \]
We have two pieces of information from the problem:
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For 5 passes and a total cost of $390: \[ 390 = 5m + b \quad (1) \]
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For 8 passes and a total cost of $615: \[ 615 = 8m + b \quad (2) \]
Now, we can solve these two equations simultaneously. Let's first solve for \(b\) in terms of \(m\) using equation (1):
From equation (1): \[ b = 390 - 5m \quad (3) \]
Next, we can substitute equation (3) into equation (2):
\[ 615 = 8m + (390 - 5m) \] \[ 615 = 8m + 390 - 5m \] \[ 615 = 3m + 390 \] Now, subtract 390 from both sides: \[ 615 - 390 = 3m \] \[ 225 = 3m \] \[ m = \frac{225}{3} = 75 \]
Now we have the cost per pass \(m = 75\). Let's use equation (3) to find \(b\):
\[ b = 390 - 5(75) \] \[ b = 390 - 375 \] \[ b = 15 \]
Now we have both \(m\) and \(b\). Therefore, the linear equation representing the total cost in dollars for \(x\) passes is:
\[ y = 75x + 15 \]
So, the correct answer is:
B. \(y = 75x + 15\)