College

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 = 5(x + 60)[/tex]

C. [tex]y + 220 = 2(x + 60)[/tex]

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

Answer :

We are given two points on the profit line:

- When [tex]$x = 60$[/tex] magazines, the profit is [tex]$y = 220$[/tex] dollars.
- When [tex]$x = 100$[/tex] magazines, the profit is [tex]$y = 420$[/tex] dollars.

Step 1. Calculate the Slope

The slope [tex]$m$[/tex] of a line through two points [tex]$\left(x_1,y_1\right)$[/tex] and [tex]$\left(x_2,y_2\right)$[/tex] is given by

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

Substitute the given values:

[tex]$$
m = \frac{420 - 220}{100 - 60} = \frac{200}{40} = 5.
$$[/tex]

Step 2. Write the Equation in Point-Slope Form

Using the point-slope form of a line,

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

we take the point [tex]$\left(60, 220\right)$[/tex] and substitute [tex]$m=5$[/tex]:

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

This is the equation that models the total profit [tex]$y$[/tex] based on the number of magazines sold [tex]$x$[/tex].

Step 3. Identify the Correct Option

Comparing with the provided options, we see that the equation

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

corresponds to option D.

Thus, the correct answer is option D.

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