Find the equation of a line with two points

Point Slope or Slope Intercept ?

There are a few different ways to write the equation of line .

Point Slope Form is better

Point slope form requires fewer steps and fewer calculations overall. This page will explore both approaches. You can click here to see a side by side comparison of the 2 forms.

In the last lesson, I showed you how to get the equation of a line given a point and a slope using the formula

Anytime we need to get the equation of a line, we need two things

a point

a slope

ALWAYS!

So, what do we do if we are just given two points and no slope?

No problem -- we'll just use the two points to pop the slope using this guy:

Check it out:

Let's find the equation of the line that passes through the points

This one's a two-stepper...

STEP 1:  Find the slope

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You can find an equation of a straight line given two points laying on that line. However, there exist different forms for a line equation. Here you can find two calculators for an equation of a line:

• first calculator finds the line equation in slope-intercept form, that is,
  

  
  It also outputs slope and intercept parameters and displays the line on a graph.

• second calculator finds the line equation in parametric form, that is,
  
  
  It also outputs a direction vector and displays line and direction vector on a graph.

Also, the text and formulas below the calculators describe how to find the equation of a line from two points manually.

Calculation precision

Digits after the decimal point: 2

Calculation precision

Digits after the decimal point: 2

How to find the equation of a line in slope-intercept form

Let's find slope-intercept form of a line equation from the two known points

and .
We need to find slope a and intercept b.
For two known points we have two equations in respect to a and b

Let's subtract the first from the second

And from there

Note that b can be expressed like this

So, once we have a, it is easy to calculate b simply by plugging or to the expression above.

Finally, we use the calculated a and b to write the result as

Equation of a vertical line

Note that in the case of a vertical line, the slope and the intercept are undefined because the line runs parallel to the y-axis. The line equation, in this case, becomes

Equation of a horizontal line

Note that in the case of a horizontal line, the slope is zero and the intercept is equal to the y-coordinate of points because the line runs parallel to the x-axis. The line equation, in this case, becomes

How to find the slope-intercept equation of a line example

Problem: Find the equation of a line in the slope-intercept form given points (-1, 1) and (2, 4)
Solution:

1. Calculate the slope a:
   
2. Calculate the intercept b using coordinates of either point. Here we use the coordinates (-1, 1):
   
3. Write the final line equation (we omit the slope, because it equals one):
   

And here is how you should enter this problem into the calculator above: slope-intercept line equation example

Parametric line equations

Let's find out parametric form of a line equation from the two known points and .
We need to find components of the direction vector also known as displacement vector.

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

to , our parametric equations will be

Note that if , then and if , then

Equation of a vertical line

Note that in the case of a vertical line, the horizontal displacement is zero because the line runs parallel to the y-axis. The line equations, in this case, become

Equation of a horizontal line

Note that in the case of a horizontal line, the vertical displacement is zero because the line runs parallel to the x-axis. The line equations, in this case, become

How to find the parametric equation of a line example

Problem: Find the equation of a line in the parametric form given points (-1, 1) and (2, 4)
Solution:

1. Calculate the displacement vector:
   
2. Write the final line equations: