91Ó°ÊÓ

To find the equation of a line, we first need to calculate the slope. The slope (denoted as \(m\)) represents the steepness and direction of a line.
You can calculate the slope between two points using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
The points given in this exercise are (2, -5) and (0, -1).
Applying these coordinates, we get:
\[ m = \frac{-1 - (-5)}{0 - 2} = \frac{4}{-2} = -2 \]
This means the slope of our line is \(-2\).

Once you've calculated the slope, you can use the point-slope form to find the equation of the line. The point-slope form is:
\[ y - y_1 = m(x - x_1) \]
Now, choose one of the points. Here, we'll use (2, -5).
By substituting the point (2, -5) and the slope \(m = -2\), we get:
\[ y - (-5) = -2(x - 2) \]
This simplifies to:
\[ y + 5 = -2(x - 2) \]
This is the point-slope form of our equation.

To get the equation in slope-intercept form, which is \(y = mx + b\), we need to simplify the point-slope form equation.
Starting from:
\[ y + 5 = -2(x - 2) \]
First, distribute the \(-2\):
\[ y + 5 = -2x + 4 \]
Next, isolate \(y\) by subtracting 5 from both sides:
\[ y = -2x + 4 - 5\]
This simplifies to:
\[ y = -2x - 1 \]
This is the slope-intercept form of the equation. Verify the equation by plugging in both given points (2, -5) and (0, -1) to ensure they satisfy the equation. Since both points work, the equation is verified correct.
Now, you understand how to transition from slope calculation to point-slope form and finally to the slope-intercept form of a linear function.