2.3 Substitution

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: replacing letters with numbers, including negative numbers, powers and fractions. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Substitute positive whole numbers into expressions
• Substitute negative numbers and decimals correctly, using brackets
• Use the order of operations with powers, brackets and fractions after substituting
• Substitute values into a formula to find the value of the subject

Key words

• Substitute: Replace each letter with its given number.
• Evaluate: Work out the value of an expression.
• Subject: The letter on its own on one side of a formula. In P = 2l + 2w the subject is P.

Substituting positive numbers

Write the expression out again with numbers in place of the letters. Put back the hidden × signs, then use BIDMAS.

Worked example: squares and brackets

Find the value of (a) 2a² and (b) (2a)² when a = 3.
(a) 2a² = 2 × 3² = 2 × 9 = 18 (only the a is squared)
(b) (2a)² = (2 × 3)² = 6² = 36

Answer: (a) 18 (b) 36

Substituting negative numbers

Put negative numbers in brackets so you do not lose the sign. Then use the rules for negative numbers.

Worked example: x = −3

(a) 2x + 7 = 2 × (−3) + 7 = −6 + 7 = 1
(b) x² = (−3)² = 9 (negative × negative is positive)
(c) 5 − x = 5 − (−3) = 8
(d) 4x² = 4 × 9 = 36

Answer: 1, 9, 8, 36

Tip: On a calculator type (−3)² with brackets. Without them you get −9.

Powers, brackets and fractions

A fraction line works like a bracket: work out the top and the bottom first, then divide.

Worked example: a = 6 and b = −2

(a) ab = 6 × (−2) = −12
(b) a² − b² = 36 − 4 = 32
(c) (a + b)/2 = (6 + (−2)) ÷ 2 = 4 ÷ 2 = 2
(d) 3(a − b) = 3 × (6 − (−2)) = 3 × 8 = 24

Answer: −12, 32, 2, 24

Substituting into formulae

A formula has a subject. Substitute the values on the right-hand side to find the subject.

Worked example: speed formula

v = u + at. Find v when u = 5, a = 3 and t = 4.
v = 5 + 3 × 4 = 5 + 12 = 17

Answer: v = 17

Worked example: perimeter of a rectangle

P = 2l + 2w. Find P when l = 8 cm and w = 3.5 cm.
P = 2 × 8 + 2 × 3.5 = 16 + 7 = 23

Answer: 23 cm

Common mistakes

• Reading 4a as 4 and a side by side: when a = 5, 4a is 20, not 45.
• Squaring the coefficient as well: 2a² with a = 3 is 18, not 36.
• Losing the negative sign: when x = −3, x² = 9, not −9.
• Working left to right instead of using BIDMAS: 5 + 3 × 4 is 17, not 32.

Revision questions

Try these, then check your answers below each one.

Question 1. Work out the value of 4a + b when a = 3 and b = 7.

Answer: 19

Working: 4 × 3 + 7 = 12 + 7 = 19.

Question 2. Work out the value of 3x + 20 when x = -5.

Answer: 5

Working: 3 × (-5) + 20 = -15 + 20 = 5.

Question 3. Work out the value of x² when x = -4.

Answer: 16

Working: (-4) × (-4) = 16.

Related lessons

• Previous lesson: 2.2 Simplifying Expressions
• Next lesson: 2.4 Formulae
• In this chapter: 2.1 Algebraic Expressions
• In this chapter: 2.5 Expanding Brackets
• In this chapter: 2.6 Factorising

Continue in the full course

This lesson is section 2.3 (EGCMF0203) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.