After the journalism club sold 60 magazines, it had [tex]$220[/tex] in profit. After it sold a total of 100 magazines, it had a total of [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 = 2(x - 60)[/tex]
C. [tex]y - 220 = 5(x - 60)[/tex]
D. [tex]y + 220 = 5(x + 60)[/tex]

To solve this problem, we need to find the equation that models the total profit, [tex]$ y $[/tex], based on the number of magazines sold, [tex]$ x $[/tex]. We have two points: (60, 220) and (100, 420). These points represent the number of magazines sold and the corresponding total profit.

Step 1: Calculate the slope (m).

The slope of a line is calculated using the formula:

[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Substitute the given points (60, 220) and (100, 420):

[tex]\[ m = \frac{420 - 220}{100 - 60} \][/tex]

[tex]\[ m = \frac{200}{40} \][/tex]

[tex]\[ m = 5 \][/tex]

Step 2: Use the point-slope form of a linear equation.

The point-slope form of a linear equation is:

[tex]\[ y - y_1 = m(x - x_1) \][/tex]

We can use one of the points, say (60, 220), and the slope we found:

[tex]\[ y - 220 = 5(x - 60) \][/tex]

Step 3: Identify the correct equation option.

Now, let's match this equation with the given options. The correct equation must be of the form:

[tex]\[ y - 220 = 5(x - 60) \][/tex]

Comparing this with the options:

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

The equation that matches what we derived is:

C. [tex]$ y - 220 = 5(x - 60) $[/tex]

Therefore, option C is the correct equation that models the total profit based on the number of magazines sold.