GCSE(F),

**Substitution** is the replacement of **variables** (letters) in an expression with numbers. An expression of 4*a* means 4 x *a*: if *a* is known to have a value of 3, then replacing *a* gives 4 x 3 (=12).

Remember that there is a hidden times sign between the the number 4 and the letter *a*.

An expression may have more than one variable. For example, 3(*a* + 2*b*) has two variables: *a* and *b*. If *a* = 4 and *b* = 5, then substituting these values into the expression:

3(*a* + 2*b*)

= 3(4 + 2 x 5) (replacing *a* with 4 and *b* with 5)

= 3(4 + 10) (multiply 2 x 5 first: BIDMAS)

= 3 x 14 = 42.

Substitution is commonly used when evaluating **formulae**. The area of a rectangle is given by *A* = *l* x *w*, where *A* is the area, *l* is the length and *w* is the width. If the length of a rectangle is 6cm, and the width is 4cm, then the area is:

*A* = *l* x *w*

*A* = 6 x 4 (6 replaces *l* and 4 replaces *w*)

*A* = 24cm^{2}

1. The calculation of tax paid for income above £11,000 is given by the formula *T* = (*i* - 11000) x 0.2 where *T* is the tax due and *i* is the income. What tax is due on an income of £23,400?

Answer: £2480

*T* = (*i* - 11000) x 0.2

*T* = (23400 - 11000) x 0.2

*T* = 12400 x 0.2

= 2480

2. The area of a square is given by *A* = *l*^{2}. If the length of a square is 4cm, what is the area of the square?

Answer: 16cm^{2}

*A* = *l*^{2}

*A* = 4^{2} (change *l* to 4)

*A* = 16

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