Two point form calculator

How to find equation of the line determined by two points?

To find equation of the line passing through points $A(x_A, y_A)$ and $B(x_B, y_B)$ ( $ x_A \ne x_B $ ), we use formula:

$$ {\color{blue}{ y - y_A = \frac{y_B - y_A}{x_B-x_A}(x-x_A) }} $$

Example:

Find the equation of the line determined by $A(-2, 4)$ and $B(3, -2)$.

Solution:

In this example we have: $ x_A = -2,~~ y_A = 4,~~ x_B = 3,~~ y_B = -2$. So we have:

$$ \begin{aligned} y - y_A & = \frac{y_B - y_A}{x_B-x_A}(x-x_A) \\ y - 4 & = \frac{-2 - 4}{3 - (-2)}(x - (-2)) \\ y - 4 & = \frac{-6}{5}(x + 2) \end{aligned} $$

Multiply both sides with $5$ to get rid of the fractions.

$$ \begin{aligned} (y - 4)\cdot {\color{red}{ 5 }} & = \frac{-6}{5}\cdot {\color{red}{ 5 }}(x + 2)\\ 5y - 20 & = -6(x + 2)\\ 5y - 20 & = -6x - 12 \\ 5y & = -6x - 12 + 20 \\ 5y & = -6x + 8 \\ {\color{blue}{ y }} & {\color{blue}{ = -\frac{6}{5}x - \frac{8}{5} }} \end{aligned} $$

In special case (when $x_A = x_B$ the equation of the line is:

$$ {\color{blue}{ x = x_A }} $$

Example 2:

Find the equation of the line determined by $A(2, 4)$ and $B(2, -1)$.

Solution:

In this example we have: $ x_A = 2,~~ y_A = 4,$ $ x_B = 2,~~ y_B = -1$. Since $x_A = x_B$, the equation of the line is:

$$ {\color{blue}{ x = 2 }} $$

You can see from picture on the right that in special case the line is parallel to y - axis.

Note: use above calculator to check the results.