To determine which equation models the total profit, \(y\), based on the number of tickets sold, \(x\), we need to analyze the given information step by step:
1. Identify the two points given:
- Point 1: \( (x_1, y_1) = (100, 300) \)
- Point 2: \( (x_2, y_2) = (200, 700) \)
2. Calculate the rate of profit per ticket (slope, \(m\)):
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{700 - 300}{200 - 100} = \frac{400}{100} = 4
\][/tex]
The slope \(m\) is 4, indicating that the profit increases by $4 for each ticket sold.
3. Use the point-slope form of the equation of a line to find the linear relationship:
The point-slope form is \( y - y_1 = m(x - x_1) \). Using point \( (100, 300) \) and slope \(m = 4\):
[tex]\[
y - 300 = 4(x - 100)
\][/tex]
Therefore, the equation that models the total profit \( y \) based on the number of tickets sold \( x \) is:
[tex]\[
\boxed{y - 300 = 4(x - 100)}
\][/tex]
Let's match this with the given multiple-choice options:
- A. \( y + 300 = 4(x + 100) \) — Incorrect form.
- B. \( y - 300 = 4(x - 100) \) — Correct form.
- C. \( y - 300 = 2.5(x-100) \) — Incorrect slope.
- D. \( y + 300 = 2.5(x+100) \) — Incorrect form and slope.
The correct equation is:
[tex]\[ \boxed{y - 300 = 4(x - 100)} \][/tex]
