How to rearrange a formula requiring factorising (harder transposition). Express h in terms of m and n.


Sometimes when you rearrange a formula you may struggle to isolate the letter that you need to make the subject. If this is the case then you probably need to partially factorise the terms by taking out the letter you need to make the subject of. It’s easier to explain this by looking at some examples. However, before you start make sure you can make subjects of easier formulas (see links at the bottom of the page).

Example 1

Make a formula for x in terms of k and y.

7y + 2xy = k – 4x

First you will need to move the terms with a x to the left hand side and any other terms without a x in to the right hand side:

7y + 2xy = k – 4x (take away 7y from both sides)

2xy = k -4x -7y (add 4x to both sides)

2xy + 4x = k – 7y

Now since the term on the left hand side cannot be combined you need to factorise the left hand side. You can do this by putting the x at the start of the bracket (there is no need to take any other factors out):

x(2y + 4) = k-7y

All you need to do now is get rid of 2y + 4 by diving both sides by 2y + 4 to give:

x = (k – 7y)/(2y+4)

So x is now the subject of the formula.

Example 2

Express h in terms of m and n.

9h – 4m = 3hn + 21n

First you will need to move the terms with a h to the left hand side and any other terms without a h to the right hand side:

9h – 4m = 3hn + 21n (take away 3hn from both sides)

9h -4m -3hn = 21n (add 4m to both sides)

9h – 3hn = 21n + 4m

Now since the term on the left hand side cannot be combined you need to factorise the left hand side. You can do this by putting the h at the start of the bracket (there is no need to take any other factors out):

h(9 – 3n) = 21n + 4m

All you need to do now is get rid of 9 – 3n by diving both sides by 9 – 3n to give:

h = (21n + 4m)/(9 – 3n)

So h is now the subject of the formula.

If you are finding these formulas difficult then you probably need to practice some easier formulas:

Rearranging a formula method. Express x in terms of y (1 inverse only)

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