Revision notes for Edexcel GCSE Maths Substitution. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Maths (1MA1) specification, so the content matches what's examinable rather than general Maths background.
Substitution
What you'll learn
- What substitution means in algebra.
- How to replace letters with numbers safely.
- How to use BIDMAS after substituting.
- How to handle negatives, powers and fractions.
1. What substitution means
Algebra uses letters to stand for numbers. In substitution questions, the question gives you the number for each letter. Your job is to put those numbers into the expression or formula, then calculate.
Definition
Key words
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A variable is a letter that stands for a number.
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An expression is a calculation with numbers and variables, such as 4x+5y4x + 5y4x+5y.
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A formula is a rule that connects variables, such as A=lwA = lwA=lw.
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Substitution means replacing each variable with its given number, then calculating the value, which is the final number you get.
Key Idea
Main idea
Substitution is not about solving for a letter. The letter values are already given, so you replace the letters and work out the answer.
Example
Simple expression
You are told a=6a = 6a=6 and b=4b = 4b=4. Work out the value of 2a+5b2a + 5b2a+5b.
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Read 2a2a2a as 2×a2 \times a2×a and 5b5b5b as 5×b5 \times b5×b.
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Replace aaa with 6 and bbb with 4.
2a+5b=2(6)+5(4)2a + 5b = 2(6) + 5(4)2a+5b=2(6)+5(4) -
Multiply first.
2(6)+5(4)=12+202(6) + 5(4) = 12 + 202(6)+5(4)=12+20 -
Add the two parts.
12+20=3212 + 20 = 3212+20=32
Tip
Hidden multiplication
When a number and a letter touch, they are being multiplied: 7m7m7m means 7×m7 \times m7×m. When two letters touch, like jkjkjk, that means j×kj \times kj×k.
2. Substitute, then use BIDMAS
Once the numbers are in place, you must calculate in the correct order.
Definition
BIDMAS
BIDMAS tells you the order of operations: Brackets first, then Indices, then Division and Multiplication, then Addition and Subtraction. Indices are powers, such as the 2 in x2x^2x2.
Brackets group part of a calculation together. If there are brackets, work inside them first.
Example
Formula with brackets
A formula is T=4(k+3)−5T = 4(k + 3) - 5T=4(k+3)−5. Find TTT when k=6k = 6k=6.
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Write the formula and replace kkk with 6.
T=4(6+3)−5T = 4(6 + 3) - 5T=4(6+3)−5 -
Work inside the bracket first.
T=4(9)−5T = 4(9) - 5T=4(9)−5 -
Multiply before subtracting.
T=36−5T = 36 - 5T=36−5 -
Subtract.
T=31T = 31T=31
3. Substituting negative numbers
A negative number is a number below zero, like -5. Negative values need extra care because the minus sign belongs to the number.
The safest method is to put negative substituted values in brackets.
Example
Negative values
You are told p=−5p = -5p=−5 and r=−4r = -4r=−4. Work out the value of 3p−2r3p - 2r3p−2r.
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Substitute using brackets around the negative values.
3p−2r=3(−5)−2(−4)3p - 2r = 3(-5) - 2(-4)3p−2r=3(−5)−2(−4) -
Work out each multiplication.
3(−5)=−15and2(−4)=−83(-5) = -15 \quad \text{and} \quad 2(-4) = -83(−5)=−15and2(−4)=−8 -
Put these results into the expression.
3p−2r=−15−(−8)3p - 2r = -15 - (-8)3p−2r=−15−(−8) -
Subtracting a negative means adding.
−15−(−8)=−7-15 - (-8) = -7−15−(−8)=−7
Common Mistake
Losing the negative sign
If a value is negative, substitute the whole number in brackets, such as writing 3(−5)3(-5)3(−5) instead of trying to remember the minus sign later.
4. Letters touching and powers
Sometimes you will see letters next to each other, such as acacac. This means multiply: ac=a×cac = a \times cac=a×c.
You may also see powers, such as b2b^2b2. This means b×bb \times bb×b. If bbb is negative, the brackets really matter.
Example
Products and powers together
You are told a=2a = 2a=2, b=−5b = -5b=−5 and c=4c = 4c=4. Work out b2−3acb^2 - 3acb2−3ac.
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Replace the letters with their values. Put the negative value in brackets.
b2−3ac=(−5)2−3(2)(4)b^2 - 3ac = (-5)^2 - 3(2)(4)b2−3ac=(−5)2−3(2)(4) -
Work out the power first.
(−5)2=25(-5)^2 = 25(−5)2=25 -
Work out the multiplication.
3(2)(4)=243(2)(4) = 243(2)(4)=24 -
Subtract.
25−24=125 - 24 = 125−24=1
Tip
Power check
A negative number squared becomes positive because a negative times a negative is positive.
5. Fractions and longer formulae
Some formulae include fractions. The fraction line means divide. For a bigger fraction, work out the top and the bottom separately before dividing.
Example
Longer formula with a fraction
A formula is R=p2−q22rR = \frac{p^2 - q^2}{2r}R=2rp2−q2. Find RRR when p=9p = 9p=9, q=3q = 3q=3 and r=4r = 4r=4.
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Substitute the values carefully.
R=92−322(4)R = \frac{9^2 - 3^2}{2(4)}R=2(4)92−32 -
Work out the powers on the top.
R=81−92(4)R = \frac{81 - 9}{2(4)}R=2(4)81−9 -
Work out the top and the bottom.
R=728R = \frac{72}{8}R=872 -
Divide.
R=9R = 9R=9
Exam technique
In the exam
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Write a clear substitution line before calculating.
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Use brackets when substituting negative numbers.
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Follow BIDMAS carefully, especially with brackets, powers and fractions.
Self review
Check yourself
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What does 4x4x4x mean if x=−3x = -3x=−3?
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In 2(a+b)2(a + b)2(a+b), what should you work out first?
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If y=−6y = -6y=−6, is y2y^2y2 positive or negative?
Recap questions
Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.
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