To find the equation of the line that passes through the points \((7.5, -2)\) and \((-5, 8)\) in slope-intercept form (which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept), we first need to calculate the slope (\(m\)).
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \((x_1, y_1) = (7.5, -2)\) and \((x_2, y_2) = (-5, 8)\).
Calculating the slope:
\[ m = \frac{8 - (-2)}{-5 - 7.5} = \frac{8 + 2}{-5 - 7.5} = \frac{10}{-12.5} = -\frac{4}{5} \]
Now we have the slope \(m = -\frac{4}{5}\).
Next, we can use either point to find the y-intercept \(b\) using the slope-intercept form \(y = mx + b\). Using the point \((7.5, -2)\):
\[ -2 = -\frac{4}{5}(7.5) + b \]
Calculating \(-\frac{4}{5}(7.5)\):
\[ -\frac{4}{5} \cdot 7.5 = -6 \]
So we have:
\[ -2 = -6 + b \]
Solving for \(b\):
\[ b = -2 + 6 = 4 \]
Now we can write the equation of the line:
\[ y = -\frac{4}{5}x + 4 \]
From the given options, the correct response that matches this equation (in slope-intercept form) is:
\[ y = -\frac{4}{5}x + 4 \]