# Manipulating more difficult formulae

**Formulae where the subject appears twice**

Where it is required to make the subject of a variable, there are times when that variable appear twice. For example, when making *w* the subject, the following steps are required:

*S* = 2*lw* + 2*lh* + 2*hw*

rearrange the formula to collect all terms containing *w* on the same side of the equals sign. *w *is then isolated by factorizing.

2*lw* + 2*hw *= *S – 2lh *(substract 2*lh* from both sides)

w(2*l* + 2*h)** *= *S – 2lh *(factorise *w, *only isolate the* w *and not the 2)

* *

**Formulae involving fractions**

Formulas which include fractions are multiplied out first before changing the subject. For example:

(note: *R* is already on the left hand side)

* * (multiply all by *Tab*)

*ab* = *Rb* + *Ra *(simplify the formula by removing the common denominator and numerator)

*ab* = *R*(*b* + *a*) (factorize)

(divide both sides by *a* + *b*

**Formulae involving powers and roots**

To make the *a* in the following formula as the subject, the following steps will be required:

*c*² = *a*² + *b*² (square both sides)

*c*² – *b*² = *a*² (subtract *b*² from both sides)

(square root both sides)