To find the equation of the line that passes through the points (5, -4) and (-1, 8), the first step is to calculate the slope of the line using the two given points. The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let’s assign the points:
- Point 1: \( (x_1, y_1) = (5, -4) \)
- Point 2: \( (x_2, y_2) = (-1, 8) \)
Substituting the values into the formula, we have:
\[ m = \frac{8 - (-4)}{-1 - 5} \]
Calculating this gives:
\[ m = \frac{8 + 4}{-1 - 5} = \frac{12}{-6} = -2 \]
So the slope \( m = -2 \).
Now, the next step would be to use the point-slope form of the linear equation, which is given by:
\[ y - y_1 = m(x - x_1) \]
You can use either of the points, but let’s use \( (5, -4) \):
\[ y - (-4) = -2(x - 5) \]
Simplifying this, we get:
\[ y + 4 = -2(x - 5) \] \[ y + 4 = -2x + 10 \] \[ y = -2x + 10 - 4 \] \[ y = -2x + 6 \]
Thus, the equation of the line in slope-intercept form is:
\[ y = -2x + 6 \]
From here, you could check specific points regarding their intersections with other lines, but calculating the slope was the essential first step in this problem.