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.

Answers

GPT-4o mini
To find the equation of the line that passes through the points (1, –5) and (–3, 7), we'll start by calculating the slope and then write the equation in the requested forms.

### a. Point-Slope Form

1. **Calculate the slope (m)** using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, let \((x_1, y_1) = (1, -5)\) and \((x_2, y_2) = (-3, 7)\).
\[
m = \frac{7 - (-5)}{-3 - 1} = \frac{7 + 5}{-4} = \frac{12}{-4} = -3
\]

2. **Using one of the points and the slope, write the point-slope form**:
The point-slope form of the equation of a line is:
\[
y - y_1 = m(x - x_1)
\]
Using point (1, -5):
\[
y - (-5) = -3(x - 1)
\]
Simplifying this, we get:
\[
y + 5 = -3(x - 1)
\]
So the equation in point-slope form is:
\[
y + 5 = -3(x - 1)
\]

### b. Slope-Intercept Form

3. **Rearranging the point-slope form to slope-intercept form (y = mx + b)**:
Start from the point-slope form:
\[
y + 5 = -3(x - 1)
\]

Distributing on the right side:
\[
y + 5 = -3x + 3
\]

Now, subtract 5 from both sides:
\[
y = -3x + 3 - 5
\]
\[
y = -3x - 2
\]

So, the final equations are:

**Point-Slope Form:**
\[
y + 5 = -3(x - 1)
\]

**Slope-Intercept Form:**
\[
y = -3x - 2
\]

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