Understanding Engineering Mathematics

No simple cancellation jumps out at us here. If we want to eliminatey,

say, then the easiest way is to multiply (i) by 2 to get

4 x− 2 y= 2

and add this result to (ii) to obtain

4 x− 2 y+x+ 2 y= 5 x= 2 + 2 = 4

Sox= 4 /5.

Now substitute back in (i) to findy

y= 2 x− 1 =^85 − 1 =^35

The solution is therefore

x=^45 ,y=^35

2.2.5 Equalities and identities

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You might not have noticed but so far we have used the equals sign,=, in two distinct
contexts. Look at the two ‘equations’:

3 x+ 6 = 12 (i)

(x− 1 )(x+ 1 )=x^2 − 1 (ii)

The first is the usual sort of equation. It only holds for a particular value ofx–i.e.x=6.
The equality in (i) enables us to determinex.
The second ‘equation’ actually tells us nothing aboutx–it is true foranyvalue of
x. We call this anidentity. The expressions on either side of the equals sign are merely
alternative forms of each other. To distinguish suchidentitiesfrom ordinary equalities we
use the symbol≡(a ‘stronger’ form of equals!) and write

(x− 1 )(x+ 1 )≡x^2 − 1

≡is read as ‘is equivalent to’, or ‘is identical to’.
The powerful thing about an identity inxis that it must be true forall values ofx.
We can sometimes use this to gain useful information (a particularly important application
occurs in partial fractions – see Section 2.2.10).

Example

Suppose we are given

x^2 − 3 x+ 2 ≡Ax^2 +Bx+C