How to Find the Equation of a Line
You can determine the equation of any straight line if you know two points on that line.
Step-by-Step Method
- Find the Slope (m): Use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
- Use Point-Slope Form: Substitute the slope and one point $(x_1, y_1)$ into $y - y_1 = m(x - x_1)$.
- Simplify: Rearrange into Slope-Intercept form ($y = mx + b$) or Standard form ($Ax + By = C$).
Formulas
Slope: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Point-Slope Form: $$ y - y_1 = m(x - x_1) $$
Slope-Intercept Form: $$ y = mx + b $$
Point-Slope Form: $$ y - y_1 = m(x - x_1) $$
Slope-Intercept Form: $$ y = mx + b $$
Example Problem
Question: Find the equation of the line passing through A(1, 2) and B(3, 6).
Step 1: Find Slope (m).
$$ m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 $$
Step 2: Use Point-Slope Form with (1, 2).
$y - 2 = 2(x - 1)$
Step 3: Convert to Slope-Intercept Form.
$y - 2 = 2x - 2$
$y = 2x - 2 + 2$
$y = 2x$
Answer: The equation is y = 2x (or $2x - y = 0$).
Frequently Asked Questions
What is the "b" in y = mx + b?
It represents the y-intercept, which is the point where the line crosses the vertical y-axis.
How do I handle vertical lines?
If $x_1 = x_2$, the slope is undefined. The equation is simply $x = x_1$.
How do I handle horizontal lines?
If $y_1 = y_2$, the slope is 0. The equation is simply $y = y_1$.