Understanding Engineering Mathematics

(iv) For (−1,−1), (−2,−3), the gradient is

m=

− 3 + 1

− 2 + 1

= 2

7.2.4 Equation of a straight line

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204 221➤

Knowing that thegradientof the straight-line segmentABis

m=

AC

BC

=

y 2 −y 1

x 2 −x 1

we can now derive an equation for the straight line through the two pointsA,Bby
equating its gradient tom. For a general pointP(x,y)on the line, its gradient is given
by(y−y 1 )/(x−x 1 ), say, which also givesm:

y−y 1

x−x 1

=

y 2 −y 1

x 2 −x 1

=m

Rearranging this (see RE7.3.4C) gives the equation of the straight line as

y−y 1 =

(

y 2 −y 1

x 2 −x 1

)

(x−x 1 )

or
y−y 1 =m(x−x 1 )

We can also write this in the formy=mx+y 1 −mx 1 and so the general form of the
equation of a straight line can be written as:

y=mx+c

wheremis the gradient of the line, andcis theintercept on they-axis. This is illustrated
in Figure 7.7.

B

A

P

y

x

(0, c), y - intercept

q

Gradient m = tan q

Figure 7.7Gradient and intercept of a liney=mx+c.