Understanding Engineering Mathematics

so

c=^32

and the equation isy=^12 ( 3 −x)orx+ 2 y− 3 =0.

7.2.6 Intersecting lines

➤

205 222➤

If two lines intersect, then their point of intersection must satisfy both of their equations.
Thus, if the lines are:

ax+by=c

dx+ey=f

then at the point of intersection these equations must be satisfied simultaneously (48

➤
).
Although one would find this intersection point by solving these equations, the geomet-
rical picture provides a nice visual means of discussing the different possibilities for the
solution of the equations. Thus, we find the following cases, with some examples you can
check:

• if the lines are not parallel (i.e. do not have the same gradient) then they
  will intersect at some point – i.e. the equations will have a solution.

Example

x+ 2 y= 1

x−y= 0

have different gradients and intersect at

( 1

3 ,

1

3

)

.

• If the lines are parallel (same gradient) then they will never intersect
  and there will be no solution to the equations.

Example

2 x+y= 0

4 x+ 2 y= 3

These are parallel, but through different points – the first passes through

the origin, the second through

(

0 ,^32

)

. These lines will never intersect
and so the equations have no solution.

• If the lines areidentical, i.e. coincide (same gradients, same intercepts),
  then they intersect at every point on the line(s) – the equations have an
  infinite number of solutions.

Example

x+ 3 y= 1

2 x+ 6 y= 2