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Line equation from two points

This online calculator finds the equation of a line given two points it passes through, in slope-intercept and parametric forms

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These online calculators find the equation of a line from 2 points.
First calculator finds the line equation in slope-intercept form, that is, y=ax+b. It also outputs slope and intercept parameters and displays line on a graph.
Second calculator finds the line equation in parametric form, that is, x=at+x_0\\y=bt+y_0. It also outputs direction vector and displays line and direction vector on a graph.

A bit of theory can be found below the calculators.

PLANETCALC, Slope-intercept line equation from 2 points

Slope-intercept line equation from 2 points

First Point

Second point

Line equation
 
Slope
 
Intercept
 
Digits after the decimal point: 2



PLANETCALC, Parametric line equation from 2 points

Parametric line equation from 2 points

First Point

Second point

Equation for x
 
Equation for y
 
Direction vector
 
Digits after the decimal point: 2

Slope-intercept line equation

Let's find slope-intercept form of line equation from the two known points (x_0, y_0) and (x_1, y_1).
We need to find slope a and intercept b.
For two known points we have two equations in respect to a and b
y_0=ax_0+b\\y_1=ax_1+b

Let's subtract the first from the second
y_1 - y_0=ax_1 - ax_0+b - b\\y_1 - y_0=ax_1 - ax_0\\y_1 - y_0=a(x_1 -x_0)
And from there
a=\frac{y_1 - y_0}{x_1 -x_0}

Note that b can be expressed like this
b=y-ax
So, once we have a, it is easy to calculate b simply by plugging x_0, y_0, a or x_1, y_1, a to the expression above.

Parametric line equations

Let's find out parametric form of line equation from the two known points (x_0, y_0) and (x_1, y_1).
We need to find components of the direction vector also known as displacement vector.
D=\begin{vmatrix}d_1\\d_2\end{vmatrix}=\begin{vmatrix}x_1-x_0\\y_1-y_0\end{vmatrix}
This vector quantifies the distance and direction of an imaginary motion along a straight line from the first point to the second point.

Once we have direction vector from x_0, y_0 to x_1, y_1, our parametric equations will be
x=d_1t+x_0\\y=d_2t+y_0
Note that if t = 0, then x = x_0, y = y_0 and if t = 1, then x = x_1, y = y_1

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