Question

A line passes through (2, –1) and (8, 4).
a. Write an equation for the line in point-slope form.
b. Rewrite the equation in standard form using integers.

Answer 1

To find the equation of the line that passes through the points \((2, -1)\) and \((8, 4)\), let's follow the steps:

### a. Point-Slope Form

1. **Find the slope (m)** using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \((x_1, y_1) = (2, -1)\) and \((x_2, y_2) = (8, 4)\).

Plugging in the values:
\[
m = \frac{4 - (-1)}{8 - 2} = \frac{4 + 1}{6} = \frac{5}{6}
\]

2. **Use the point-slope form** of the equation, which is:
\[
y - y_1 = m(x - x_1)
\]
Using point \((2, -1)\) for \( (x_1, y_1) \):
\[
y - (-1) = \frac{5}{6}(x - 2)
\]
This simplifies to:
\[
y + 1 = \frac{5}{6}(x - 2)
\]

### b. Standard Form

1. **Distribute the slope (m)** on the right side:
\[
y + 1 = \frac{5}{6}x - \frac{5}{3}
\]

2. **Multiply through by 6** to eliminate the fraction:
\[
6(y + 1) = 5x - 10
\]
This simplifies to:
\[
6y + 6 = 5x - 10
\]

3. **Rearranging to standard form** \(Ax + By = C\):
\[
-5x + 6y + 16 = 0
\]
Or, rearranging terms:
\[
5x - 6y = 16
\]

### Final Answers

- **Point-Slope Form**:
\[
y + 1 = \frac{5}{6}(x - 2)
\]

- **Standard Form**:
\[
5x - 6y = 16
\]