After the journalism club sold 60 magazines, it had [tex]$\$[/tex]220[tex]$ in profit. After it sold a total of 100 magazines, it had a total of $[/tex]\[tex]$420$[/tex] in profit. Which equation models the total profit, [tex]$y$[/tex], based on the number of magazines sold, [tex]$x$[/tex]?

A. [tex]$y + 220 = 2(x + 60)$[/tex]
B. [tex]$y - 220 = 5(x - 60)$[/tex]
C. [tex]$y - 220 = 2(x - 60)$[/tex]
D. [tex]$y + 220 = 5(x + 60)$[/tex]



Answer :

To determine the correct equation modeling the total profit, \( y \), based on the number of magazines sold, \( x \), given two points \((x_1, y_1)\) = (60, 220) and \((x_2, y_2)\) = (100, 420), follow these steps:

1. Calculate the slope (rate of change of profit per magazine) \( m \):
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the given points:
[tex]\[ m = \frac{420 - 220}{100 - 60} = \frac{200}{40} = 5 \][/tex]
So, the slope \( m \) is 5.

2. Determine the equation in point-slope form:
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using the point (60, 220) and the slope 5:
[tex]\[ y - 220 = 5(x - 60) \][/tex]

3. Identify the correct equation from the given choices:
The correct equation matches the point-slope form we derived. Comparing:
[tex]\[ y - 220 = 5(x - 60) \][/tex]
We find that this matches choice B.

Thus, the equation that models the total profit \( y \) based on the number of magazines sold \( x \) is:
[tex]\[ \boxed{y - 220 = 5(x - 60)} \][/tex]

The correct answer is:
B. [tex]\( y - 220 = 5(x - 60) \)[/tex]