Understanding Engineering Mathematics

2.2.4 Simultaneous equations

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We will say more about solving equations in Section 13.5, but here we need to cover some
simple examples that we will use in subsequent sections. A simple linear equation in one
variable, of the form

ax+b= 0 (a  = 0 )

wherea,bare given constants, is easy to solve:

ax=−band sox=−

b

a

Often we have to deal with equations involving two variables, such as

ax+by=e

cx+dy=f

wherea,b,c,d,e,f are all given constants. This is referred to as asystem of simulta-
neous linear equations in the two variablesxandy. Such a system can be solved by
‘eliminating’ one of the variables, sayy, and hence determining the other.

Example
x−y= 1
x+ 2 y=− 1

In this case it is actually easier to eliminatexfirst, because we notice that by subtracting
the equations (i.e. doing the same subtraction on each side of the equation) we obtain

(x−y)−(x+ 2 y)= 1 −(− 1 )

x−y−x− 2 y= 1 + 1 = 2

So − 3 y= 2

and therefore y=−^23

We can now obtainxfrom the first equation:

x= 1 +y= 1 −^23 =^13

So the solution is

x=^13 ,y=−^23

which you should check in the original equations. Note that the answers are left as fractions
rather than converting to decimals. Any decimal form of the solutions is likely to be an
approximation and therefore incur errors, which may have serious consequences in an
actual engineering application.