Substitution means replacing a letter in an expression or formula with a given number. The number may be a positive or negative whole number, a fraction or a decimal. Substitution is used to evaluate expressions and calculate values in formulae.

 

When substituting, the given value must be placed everywhere the variable appears. Brackets should always be used when substituting negative numbers or fractions to avoid errors.

 

 

Substituting whole numbers

Consider the expression:

$$
3x + 4
$$

 

If \( x = 5 \), substitute \( 5 \) for \( x \):

$$
3(5) + 4
$$

$$
= 15 + 4
$$

$$
= 19
$$

 

 

Substituting negative numbers

If \( x = -2 \), brackets are essential:

$$
3(-2) + 4
$$

$$
= -6 + 4
$$

$$
= -2
$$

Without brackets, the calculation could be misinterpreted.

 

 

Substituting fractions

Consider the expression:

$$
4x - 1
$$

 

If \( x = \frac{3}{4} \), substitute carefully using brackets:

$$
4\left(\frac{3}{4}\right) - 1
$$

$$
= 3 - 1
$$

$$
= 2
$$

 

 

Substituting decimals

Consider the expression:

$$
2x^2
$$

 

If \( x = 1.5 \), substitute and evaluate step by step:

$$
2(1.5)^2
$$

$$
= 2 \times 2.25
$$

$$
= 4.5
$$

 

 

Substituting into formulae written in words

Sometimes a formula is given in words. For example:

 

The cost, \( C \), is found by multiplying the number of items, \( n \), by 2.5 and then adding 3.

 

This can be written as:

$$
C = 2.5n + 3
$$

 

If \( n = 4 \):

$$
C = 2.5(4) + 3
$$

$$
= 10 + 3
$$

$$
= 13
$$

 

When substituting, always:
• use brackets
• follow the correct order of operations
• check signs carefully

 

Accurate substitution is essential for evaluating expressions correctly and for solving problems involving formulae.