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Equation of a Straight Line

The equation of a straight line is usually written this way:

y = mx + b

(or "y = mx + c" in the UK see below)

 

What does it stand for?

 
Equation of a Straight Line
Slope (or Gradient) Y Intercept

 

y = how far up

x = how far along

m = Slope or Gradient (how steep the line is)

b = the Y Intercept (where the line crosses the Y axis)

 

How do you find "m" and "b"?

  • b is easy: just see where the line crosses the Y axis.
  • m (the Slope) needs some calculation:
  m  = 
Change in Y
Change in X
  gradient



Knowing this we can work out the equation of a straight line:

Example 1



m  = 
2
1
 =  2

b = 1 (where the line crosses the Y-Axis)

Therefore y = 2x + 1

Now you can choose any value for x and find the matching value for y

For example, when x is 1:

y = 2×1 + 1 = 3

Check for yourself that x=1 and y=3 is actually on the line.

Or we could choose another value for x, such as 7:

y = 2×7 + 1 = 15

And so when x=7 you will have y=15

Example 2

m  = 
3
-1
 =  –3

b = 0

This gives us y = –3x + 0
We do not need the zero!

Therefore y = –3x

 

Example 3: Vertical Line

graph x=2

What is the equation for a vertical line?
The slope is undefined ... and where does it cross the Y-Axis?

In fact, this is a special case, and you use a different equation, not "y=...", but instead you use "x=...", like this:

x = 1.5

Every point on the line has x coordinate 1.5,
that’s why its equation is x = 1.5

Rise and Run

Sometimes the words "rise" and "run" are used.

  • Rise is how far up
  • Run is how far along

And so the slope "m" is:

m  = 
rise
run

You might find that easier to remember

   rise and run

 

Now Play With The Graph !

You can see the effect of different values of m (the slope) and b (the y intercept) at Explore the Straight Line Graph

   Animation

 

Question 1 Question 2

Other Forms

What we have been looking at is called the "slope-intercept" form.

The equation of a straight line can be written in other ways. Another popular form is:

y - y1 = m(x - x1)

And you can read more about it at Point-Slope Equation of a Straight Line

Footnotes

Country Note:

Different Countries teach different "notation" (as sent to me by kind readers):

In the US and Canada the notation is: y = mx + b
In the UK, Australia, Bahamas, Brunei, India, Indonesia, Ireland, Jamaica, Malaysia, Malta, Nepal, New Zealand, Pakistan, Singapore, Solomon Islands, South Africa, and Zimbabwe y = mx + c
In Brazil, Denmark, Holland and Viet Nam: y = ax + b
In China, Finland and Ukraine: y = kx + b
In Greece: ψ = αχ + β
In Italy: y = mx + q
In Japan: y = mx + d
In Latvia: y = jx + t
In Romania: y = gA + C
In Sweden: y = kx + m
   
In your country: let us know!

... but it all means the same thing, just different letters.

 

Question 1 Question 2
Slope (Gradient) of a Straight Line Y Intercept of a Straight Line Test Yourself Explore the Straight Line Graph Straight Line Graph Calculator Graph Menu