Question

4.
A line passes through (3, –2) and (6, 2).
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 (3, -2) and (6, 2), we can follow these steps:

### a. Write an equation for the line in point-slope form.

1. **Calculate the slope (m)** using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points (3, -2) and (6, 2):
\[
m = \frac{2 - (-2)}{6 - 3} = \frac{2 + 2}{3} = \frac{4}{3}
\]

2. **Use the point-slope form of a line**, which is:
\[
y - y_1 = m(x - x_1)
\]
Using one of the points, say (3, -2):
\[
y - (-2) = \frac{4}{3}(x - 3)
\]
This simplifies to:
\[
y + 2 = \frac{4}{3}(x - 3)
\]

Thus, the equation in point-slope form is:
\[
y + 2 = \frac{4}{3}(x - 3)
\]

### b. Rewrite the equation in standard form using integers.

1. **Distribute the slope on the right side**:
\[
y + 2 = \frac{4}{3}x - 4
\]

2. **Clear the fraction by multiplying the entire equation by 3** (the denominator):
\[
3(y + 2) = 4x - 12
\]
This gives:
\[
3y + 6 = 4x - 12
\]

3. **Rearranging to standard form \(Ax + By = C\)**:
\[
-4x + 3y + 6 + 12 = 0\ \Rightarrow\ -4x + 3y + 18 = 0
\]
To put it in the standard form:
\[
4x - 3y = 18
\]

Thus, the standard form of the equation is:
\[
4x - 3y = 18
\]

### Summary:
- **Point-Slope Form:** \( y + 2 = \frac{4}{3}(x - 3) \)
- **Standard Form:** \( 4x - 3y = 18 \)