Two point form calculator

This online calculator can find and plot the equation of a straight line passing through the two points. The calculator will generate a step-by-step explanation on how to obtain the result.

Result:

The equation of the line passing through point $ A=\left(~0~,~0~\right) $ , and point $ B=\left(~100~,~100~\right) $ is:

$$ y = x $$

Explanation:

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

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

In this example we have $ x_A = 0 $ , $ y_A = 0 $ , $ x_B = 100 $ and $ y_ B = 100 $. So,

$$ \begin{aligned} y - y_A~&=~\frac{y_B - y_A}{x_B - x_A}(x - x_A) \\ y - 0~&=~\frac{ 100 - 0 }{ 100 - 0 } \left( x - 0 \right) \\y - 0 ~&=~ 1 \left( x - 0 \right) \\y - 0 ~&=~ 1x + 0 \\y ~&=~ 1x + 0 \end{aligned} $$

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Script name : two-point-form-calculator

Form values: 0 , 0 , 100 , 100 , slop , expl

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Line through two points
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General form (default)
Slope y-intercept form
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examples
example 1:ex 1:
Determine the equation of a line passing through the points $(-2, 5)$ and $(4, -2)$.
example 2:ex 2:
Find the slope - intercept form of a straight line passing through the points $\left( \frac{7}{2}, 4 \right)$ and $\left(\frac{1}{2}, 1 \right)$.
example 3:ex 3:
If points $\left( 3, -5 \right)$ and $\left(-5, -1\right)$ are lying on a straight line, determine the slope-intercept form of the line.

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:

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

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:

two point form 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.

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