Understanding Engineering Mathematics

Complications can arise with such systems of equations, depending on the coefficients
a,b,c,d,e,f. For example consider the system

x+y= 1

2 x+ 2 y= 3

This system has no solutions forxandybecause whereas the left-hand sides are related by
a factor of two, the right-hand sides are not – we say that the equations areincompatible.
To see the nonsense they lead to, subtract twice the first from the second to give

‘0= 2 x+ 2 y− 2 (x+y)= 3 − 2 = 1

Therefore 0=1’

An uncomfortable conclusion!
Now consider the pair

x+y= 1

2 x+ 2 y= 2

These are not really two different equations – the second is just twice the first. In this case
we can find any number of solutions to the system – just fix a value for, say,yand then
determine the required value ofx.
The following is an example of another important type of system

x+y= 0

2 x+y= 0

Because of the zeros on the right-hand side such a system is said to behomogeneous.
We will have a lot to say about them in Chapter 13, but for now convince yourself that
the only possible solution of this system isx=y=0, a so-called ‘trivial’ solution. In
general, show that for the system

ax+by= 0

cx+dy= 0

to have a ‘non-trivial’ solution (i.e.x,ynot both zero) for non-zero coefficientsa,b,c,
dwe must havead=bc.
The sorts of complications we have pointed to are really the concern of more advanced
mathematics. For the moment all we need to be able to do is solve a given system by elimi-
nation, when no complications arise. Graphically, equations of the formax+by=crepre-
sent straight lines in a plane and systems of simultaneous equations of this type represent
sets of such lines that may or may not intersect. We will say more about this in Chapter 7.

Solution to review question 2.1.4

2 x−y= 1 (i)

x+ 2 y= 2 (ii)