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- Two point form

This online calculator finds and plots the equation of a straight line passing through the two given points. The calculator generates a step-by-step explanation of how to get the result.

working...

Examples

ex 1:

Determine the equation of a line passing through the points $(-2, 5)$ and $(4, -2)$.

ex 2:

Find the slope - intercept form of a straight line passing through the points $\left( \frac{7}{2}, 4
\right)$
and $\left(\frac{1}{2}, 1 \right)$.

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.

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Find more worked-out examples in our database of solved problems..

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:

**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.

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