Everything Maths Grade 11

16.5 CHAPTER 16. GEOMETRY

16.5 Co-ordinate Geometry EMBCK

Equation of a Line Between Two Points EMBCL

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There are many different methods of specifyingthe requirements for determining the equationof a

straight line. One optionis to find the equation of a straight line, when two points are given.

Assume that the two points are(x 1 ; y 1 ) and(x 2 ; y 2 ), and we know that the general form of the equation

for a straight line is:

y = mx + c (16.1)

So, to determine the equation of the line passingthrough our two points,we need to determine values

for m (the gradient of the line) and c (the y-intercept of the line). The resulting equation is

y− y 1 = m(x− x 1 ) (16.2)

where (x 1 ; y 1 ) are the co-ordinates ofeither given point.

Tip

If you are asked to cal-
culate the equation of a
line passing through two
points, use:

m=
y 2 −y 1
x 2 −x 1
to calculatem and then
use:

y−y 1 =m(x−x 1 )

to determine the equa-
tion.

Extension: Finding the second equation for a straight line

This is an example of aset of simultaneous equations, because we canwrite:

y 1 = mx 1 + c (16.3)

y 2 = mx 2 + c (16.4)

We now have two equations, with two unknowns, m and c.

Subtract (16.3) from (16.4) y 2 − y 1 = mx 2 − mx 1 (16.5)

∴ m =

y 2 − y 1

x 2 − x 1

(16.6)

Re-arrange (16.3) to obtain c y 1 = mx 1 + c (16.7)

c = y 1 − mx 1 (16.8)

Now, to make things a bit easier to remember, substitute (16.7) into (16.1):

y = mx + c (16.9)

= mx + (y 1 − mx 1 ) (16.10)

which can be re-arranged to: y− y 1 = m(x− x 1 ) (16.11)

For example, the equation of the straight line passing through (−1;1) and (2;2) is given by first calcu-