Equation of a Straight Line

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

y = mx + c

What does it stand for?

/
Slope (or Gradient) / Y Intercept
y = how far up
x = how far along
m = Slope or Gradient (how steep the line is)
c = the Y Intercept (where the line crosses the Y axis)

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

  • c is easy: just see where the line crosses the Y axis.
  • m (the Slope) needs some calculation:

m = / Change in Y
Change in X
/

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

Example 2

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

Therefore y = –3x

Example 3: Vertical Line

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
/

How to find the rule – given two coordinates

Step 1: How to “find” the gradient

Step 2: How to “find” the y-intercept

Substitute a coordinate into the rule

Example:

a)Step 1: (2;6) (1 ;-8)

m=

m =

m= 14

Step 2: y = 14x + c

Subst (2; 6) into rule:

6 = 14(2) + c

6 = 28 + c

-22 = c

y = 14x -22

General form to find gradient and y-intercept

The coordinates of every point lying on a straight line will “fit” the equation of the line and every point not lying on the line will not fit the equation.

Lines parallel to axes

Lines parallel to x - axes have a zero gradient.

Lines parallel to y – axes :

Finding the rule from a table:

1.

x / 1 / 2 / 3 / 4 / 5 / 6
y / 7 / 9 / 11 / 13 / 15 / 17

First difference:

y-intercept:

Rule:

2.

x / 0 / 1 / 2 / 3 / 4 / 5 / 6
y / 10 / 8 / 6 / 4 / 2 / 0 / -2

First difference:

Y-intercept:

Rule:

Finding the rule from the graph:

Parallel lines:

Parallel lines have the same gradient

y= 4 x + 6Parallel lines

y= 4x -10

Perpendicular lines:

m 1 x m 2 = -1

y = -3x + 1

y = x + 5

-3 x = -1

  • Find the equation of the straight line through the point ( 1,3) and perpendicular to the line y = 2x + 3

Distance: Length of the line joining point A( x1, y1) to point B, (x2,y2)

Find the length of the line joining the points A ( 3,-5) and B ( 6,4)