Question

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


y – 5 = 3(x + 1); y = 3x + 8





y + 5 = –3(x – 1); y = –3x – 2

Answer 1

To find the equation of the line that passes through the points \((1, -5)\) and \((-3, 7)\), we first need to calculate the slope of the line.

### Step 1: Calculate the slope (m)

The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \((1, -5)\) and \((-3, 7)\):

\[
m = \frac{7 - (-5)}{-3 - 1} = \frac{7 + 5}{-3 - 1} = \frac{12}{-4} = -3
\]

### Step 2: Write the equation in point-slope form

The point-slope form of a line is given by:

\[
y - y_1 = m(x - x_1)
\]

Using the slope \(m = -3\) and one of the points, say \((1, -5)\):

\[
y - (-5) = -3(x - 1)
\]

This simplifies to:

\[
y + 5 = -3(x - 1)
\]

### Step 3: Rewrite in slope-intercept form

To convert to slope-intercept form \(y = mx + b\), we will distribute and simplify:

1. Distribute \(-3\):

\[
y + 5 = -3x + 3
\]

2. Subtract 5 from both sides:

\[
y = -3x + 3 - 5
\]

This simplifies to:

\[
y = -3x - 2
\]

### Summary

- The equation of the line in point-slope form is
\[
y + 5 = -3(x - 1)
\]
- The equation in slope-intercept form is
\[
y = -3x - 2
\]

Your given equations \(y - 5 = 3(x + 1); y = 3x + 8\) and \(y + 5 = -3(x - 1); y = -3x - 2\) contain an error with the first set. The correct final forms are as stated above.