Question

A line passes through (1, –5) and (–3, 7).
a. Write an equation for the line in point-slope form.
b. Rewrite the equation in slope-intercept form.


y + 5 = –3(x – 1); y = –3x – 2








y – 5 = 3(x + 1); y = 3x + 8

Answer 1

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: \(y = -3x - 2\)