Algebraic Manipulation

This chapter explores algebraic manipulation. It covers expanding brackets with common factors and changing the subject of a formula.

Expanding brackets

Suppose we wanted to expand and simplify;

Here we’re asked to multiply each of the terms in the left hand-side with (3x + y). This gives us;

Now we can expand the expression

…then we simplify;

Factorising with common factors

The opposite of expanding brackets is factorising. But to factorise we first take out the common factors.
Suppose we wanted to factorise the following expression;

We first find the common factor; 4 is the common factor of 8x and 12. So we have;

To find the terms that will be in the brackets we divide 8x and -12 by 4. Dividing gives us;

So the factorised expression becomes;

Here is another expression we can factorise;

Example

Here is another example for changing the subject. Suppose we wanted to make v the subject;

We get rid of the fraction first;

Then divide both sides by t and then subtract u.


Now we write it in a more easy to read form.

Example

Make x the subject;

Rearranging when x appears twice

Suppose we wanted to make x the subject in the following expression;

As we’ve done above we first get rid of the fraction. By doing that we get;

Next we expand the brackets;

We then collect like terms by putting the x terms on one side and the non x terms on the other as shown below;

We can now factorise the left hand side.

Then we divide both sides by (9y + 2) as shown below;