To solve this problem, we need to develop a linear equation that describes the number of patients seen each week at the clinic. Let’s break it down step by step.
1. Identify the given points: We are given two points in time with corresponding number of patients:
At week 5 (x = 5), the number of patients is 75 (y = 75).
At week 10 (x = 10), the number of patients is 50 (y = 50).
2. Calculate the slope (m): The slope of a line (m) is determined by the change in y divided by the change in x.
[tex]\[
m = \frac{y2 - y1}{x2 - x1} = \frac{50 - 75}{10 - 5} = \frac{-25}{5} = -5
\][/tex]
3. Determine the y-intercept (b): The y-intercept can be found by using the slope and one of the points given. We can use the point (5, 75).
[tex]\[
y = mx + b
\][/tex]
Substituting the known values:
[tex]\[
75 = -5(5) + b
\][/tex]
Simplify:
[tex]\[
75 = -25 + b
\][/tex]
Solving for b:
[tex]\[
b = 100
\][/tex]
4. Form the equation of the line: Using the slope (m) and y-intercept (b), the linear equation can be written in the form [tex]\( f(x) = mx + b \)[/tex].
[tex]\[
f(x) = -5x + 100
\][/tex]
Therefore, the correct equation that describes the number of patients seen each week at the clinic is:
[tex]\[
f(x) = -5x + 100
\][/tex]
So, the correct answer is:
[tex]\[ f(x) = -5x + 100 \][/tex]
