To substitute, replace each letter with its value and work out the result using the order of operations. Take particular care with powers and negative numbers: 2x² means square x first, then double, while (2x)² means double first, then square. Put negative values in brackets, so that x² with x = −3 becomes (−3)² = 9.
Substitution tips
- Replace each letter with its value, in brackets
- Follow BIDMAS: powers before multiplying, multiplying before adding
- 2x² = 2 × x²; (2x)² = (2 × x)²
- (−3)² = 9, but −3² = −9
Substituting into an expression
Replace each letter with its number, then calculate. 3x + 2 with x = 4 becomes 3 × 4 + 2 = 14. Remember that 3x means 3 × x, so with x = 4 it is 12, not 34. Writing the substituted values out before calculating helps avoid this.
Powers and order of operations
Work out powers before multiplying. In 2x² the square applies only to x, so with x = −2 it is 2 × 4 = 8. In (2x)² the whole bracket is squared: (−4)² = 16. These are different, which is why the order of operations matters.
Negative numbers
Put negative values in brackets when you substitute. With x = −2, 7 − 3x = 7 − 3 × (−2) = 7 + 6 = 13. Subtracting a negative is the same as adding, and a negative squared is positive.
Substituting into formulae
A formula is a rule connecting quantities, written with a subject on one side, such as v = u + at. Substitute every given value, then evaluate. Include units in the answer where the context has them.
Formulae with several steps
For longer formulae, such as s = ut + ½at², work out each term separately before adding. With u = 1, t = 5 and a = 4: ut = 5 and ½at² = ½ × 4 × 25 = 50, so s = 55.
Worked examples
Example 1. Work out the value of 3x + 2 when x = 4.
- 3 × 4 + 2.
- 12 + 2.
Answer: 14.
Example 2. Work out the value of x² + 3 when x = 5.
- 5² = 25.
- 25 + 3.
Answer: 28.
Example 3. When x = −2, work out 2x² and (2x)².
- 2x² = 2 × (−2)² = 2 × 4 = 8.
- (2x)² = (−4)² = 16.
Answer: 2x² = 8 and (2x)² = 16.
Example 4. Work out 7 − 3x when x = −2.
- 7 − 3 × (−2) = 7 + 6.
Answer: 13.
Example 5. Use s = ut + ½at² to find s when u = 1, t = 5 and a = 4.
- ut = 1 × 5 = 5.
- ½at² = ½ × 4 × 25 = 50.
- s = 5 + 50.
Answer: 55.
Example 6. The formula F = 9C/5 + 32 converts Celsius to Fahrenheit. Find F when C = 25.
- 9 × 25 = 225, and 225 ÷ 5 = 45.
- 45 + 32.
Answer: 77 °F.
Common mistakes
- Writing 3x as 34 when x = 4. 3x means 3 × x = 12.
- Squaring after multiplying in 2x². Square x first, then multiply by 2.
- Not bracketing negative values. Write (−2)², which equals 4.
- Ignoring the order of operations. Powers, then multiplication, then addition.
- Forgetting units in a formula. Give the units of the subject where they are known.
How substitution is examined
Substitution questions appear early on Foundation papers for both the 1MA1 and 4MA1 specifications, and substituting into formulae is used throughout Higher tier too. Questions ask you to:
- work out the value of an expression for given values
- substitute negative numbers and values with powers
- use a formula from science, such as v = u + at
- substitute into a formula and then solve for an unknown
Practise with the notation and substitution worksheet, which includes answers, and the GCSE algebra revision engine.
Substitution relies on the order of operations, and it is the reverse of rearranging formulae.
Frequently asked questions
How do you substitute into an expression?
Replace each letter with its value, then calculate using the order of operations.
What is the difference between 2x² and (2x)²?
In 2x² only x is squared; in (2x)² the whole of 2x is squared.
How do you substitute a negative number?
Put it in brackets, so that powers and signs are handled correctly.