Question

how to find the equation of a line given two points

Answer 1

To find the equation of a line given two points, you can follow these steps:

1. **Identify the Two Points**: Let the two points be \( P_1(x_1, y_1) \) and \( P_2(x_2, y_2) \).

2. **Calculate the Slope (m)**: The slope \( m \) of the line passing through the two points is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Make sure that \( x_2 \neq x_1 \) to avoid division by zero.

3. **Use the Point-Slope Form**: With the slope calculated, you can use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
You can use either of the two points \( P_1 \) or \( P_2 \) in this formula.

4. **Convert to Slope-Intercept Form (Optional)**: If you want the equation in the slope-intercept form \( y = mx + b \), you can solve for \( y \) after substituting the values.

5. **Convert to Standard Form (Optional)**: If you need the equation in the standard form \( Ax + By = C \), you can rearrange the equation accordingly.

### Example
Let's say you have two points \( P_1(1, 2) \) and \( P_2(3, 4) \).

1. Calculate the slope:
\[
m = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1
\]

2. Use the point-slope form:
Using point \( P_1(1, 2) \):
\[
y - 2 = 1(x - 1)
\]
Simplifying this:
\[
y - 2 = x - 1 \implies y = x + 1
\]

So, the equation of the line in slope-intercept form is \( y = x + 1 \).

You can check with point \( P_2(3, 4) \) to verify:
- Substituting \( x = 3 \):
\[
y = 3 + 1 = 4 \quad \text{(which matches \( P_2 \))}
\]

That's how you can find the equation of a line given two points!