Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry

P1^

2

The equation of a straight line

The word straight means going in a constant direction, that is with fixed gradient.

This fact allows you to find the equation of a straight line from first principles.

EXAMPLE 2.3 Find the equation of the straight line with gradient 2 through the point (0, −5).

SOLUTION

Take a general point (x, y) on the line, as shown in figure 2.11. The gradient of

the line joining (0, −5) to (x, y) is given by

gradient==y +

x

y

x

–(–)

–

(^5).
0

5

Since we are told that the gradient of the line is 2, this gives

y

x

• (^5) = 2 
  ⇒y= 2 x− 5.
  Since (x, y) is a general point on the line, this holds for any point on the line and
  is therefore the equation of the line.
  The example above can easily be generalised (see page 50) to give the result that
  the equation of the line with gradient m cutting the y axis at the point (0, c) is
  y = mx + c.
  (In the example above, m is 2 and c is −5.)
  This is a well-known standard form for the equation of a straight line.
  –1^01 2 3 4 5
  –1
  –2
  –3
  –4
  –5 (0, –5)
  (^2) (x, y)
  3
  1
  4
  y
  x
  Figure 2.11