Question

A line passes through (3, –2) and (6, 2). Write an equation for the line in point-slope form.
Rewrite the equation in standard form using integers.


y – 2 = (x – 3); –4x + 3y = 18


y + 2 = (x – 3); –4x + 3y = –18


y – 3 = (x + 2); –4x + 3y = 17


y + 2 = (x + 3); –4x + 3y = –18

Answer 1

To find the equation of the line that passes through the points (3, –2) and (6, 2), we first need to calculate the slope of the line.

1. **Calculate the slope (m)**:
The formula for slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using points (3, -2) and (6, 2):
\[
m = \frac{2 - (-2)}{6 - 3} = \frac{2 + 2}{3} = \frac{4}{3}
\]

2. **Write the equation in point-slope form**:
The point-slope form of the equation of a line is:
\[
y - y_1 = m(x - x_1)
\]
Using the slope \( \frac{4}{3} \) and the point (3, -2):
\[
y - (-2) = \frac{4}{3}(x - 3)
\]
This simplifies to:
\[
y + 2 = \frac{4}{3}(x - 3)
\]

3. **Convert to standard form**:
Next, we need to convert this into standard form \( Ax + By = C \) with integers. Start by eliminating the fraction:
\[
3(y + 2) = 4(x - 3)
\]
Distributing on both sides:
\[
3y + 6 = 4x - 12
\]
Rearranging it to standard form:
\[
-4x + 3y = -18
\]
To get the standard form with a positive leading coefficient, we multiply the entire equation by -1:
\[
4x - 3y = 18
\]
Which can be rewritten as:
\[
-4x + 3y = -18
\]

So the final answer in standard form using integers is:
**−4x + 3y = -18**.

Out of your provided options, the correct answer is:
**y + 2 = \frac{4}{3}(x - 3); –4x + 3y = –18**.