Question

How to find the equation of a line given a point and a parallel line with two points

Answer 1

To find the equation of a line that is parallel to a given line and passes through a specific point, you can follow these steps:

1. **Identify the given points on the parallel line:** Assume you have two points on the parallel line, \( A(x_1, y_1) \) and \( B(x_2, y_2) \).

2. **Calculate the slope of the parallel line:** The slope \( m \) of the line through points \( A \) and \( B \) can be calculated using the formula:

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

3. **Use the point-slope form:** If you have a point \( P(x_0, y_0) \) through which the new line passes and you know the slope \( m \) of the parallel line, you can use the point-slope form of a line's equation:

\[
y - y_0 = m(x - x_0)
\]

4. **Convert to slope-intercept or standard form:** If needed, rearrange the equation into the slope-intercept form \( y = mx + b \) or standard form \( Ax + By = C \).

### Example

Let's go through a concrete example:

- Given points \( A(1, 2) \) and \( B(3, 4) \) that define the parallel line.
- Given point \( P(4, 1) \) through which the new line must pass.

1. Calculate the slope of the line through points \( A \) and \( B \):
\[
m = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1
\]

2. Use the point-slope form with point \( P(4, 1) \):
\[
y - 1 = 1(x - 4)
\]
Simplifying this gives:
\[
y - 1 = x - 4
\]
\[
y = x - 3
\]

So, the equation of the new line that is parallel to the line through points \( A \) and \( B \) and passes through the point \( P(4, 1) \) is:
\[
y = x - 3
\]