To find the equation of the line that passes through the points (1, -5) and (-3, 7), we'll start by calculating the slope of the line using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let \((x_1, y_1) = (1, -5)\) and \((x_2, y_2) = (-3, 7)\):
\[ m = \frac{7 - (-5)}{-3 - 1} = \frac{7 + 5}{-3 - 1} = \frac{12}{-4} = -3 \]
a. Point-slope form
Using the point-slope form of a line, which is:
\[ y - y_1 = m(x - x_1) \]
We can choose one of the points, say \( (1, -5) \):
\[ y - (-5) = -3(x - 1) \]
This simplifies to:
\[ y + 5 = -3(x - 1) \]
So, the equation in point-slope form is:
\[ \boxed{y + 5 = -3(x - 1)} \]
b. Slope-intercept form
Now, converting to slope-intercept form (y = mx + b):
Starting with the point-slope form:
\[ y + 5 = -3(x - 1) \]
Distributing the \(-3\):
\[ y + 5 = -3x + 3 \]
Now, isolate \(y\):
\[ y = -3x + 3 - 5 \]
\[ y = -3x - 2 \]
So, the equation in slope-intercept form is:
\[ \boxed{y = -3x - 2} \]
Summary
- Point-slope form: \(y + 5 = -3(x - 1)\)
- Slope-intercept form: \(y = -3x - 2\)