Substituting numbers into expressions and formulas — one clear step at a time.
6 lessons · 35 questions (A–G) · 20 calculation + 15 word problems · no time pressure
Substituting means swapping a letter for a number and working the value out. You do it with an expression (a set of terms with no equals sign, like 3x + 2) and with a formula (a rule that does have one, like P = 4L). Pupils most often slip by adding instead of multiplying (reading 4x as a single number), getting the order wrong, or forgetting that x2 means x × x, not x × 2.
At Improve Tuition, a qualified teacher shows a pupil exactly how to replace the letter and follow the order of operations, in small worked steps — with read-aloud, comfort spacing and a reading tint for pupils who take in maths more easily that way.
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Lesson 1
What substitution means
Aim: Swap a letter for a number, then work the value out.
Substituting means putting a number in place of a letter and then working out the value. You will see it with two things. An expression is a collection of terms with no equals sign, such as x + 8. A formula is a rule that does have an equals sign, such as P = 4L. Either way, the letter is just a box waiting for a number.
An expression (3x + 2) has no equals sign; a formula (P = 4L) does. Either way, you swap the letter for a number.
Worked example
Substitute x = 5 into the expression x + 8.
Find the letter in the expression: it is x.
Replace that x with its value, 5: the expression becomes 5 + 8.
Work it out: 5 + 8 = 13.
So x + 8 = 13 when x = 5.
Check: we swapped the letter for the number and nothing else changed. ✓
Watch out
Write the number exactly where the letter was; the rest of the expression stays the same.
A number written next to a letter means multiply: 4x means 4 × x. So to substitute, replace the letter with its value and multiply. If x = 3, then 4x = 4 × 3 = 12.
5x means 5 × x. Feed x = 4 into the machine: 4 × 5 = 20.
Worked example
Substitute x = 4 into 5x.
5x means 5 × x.
Replace x with 4: 5 × 4.
Work it out: 5 × 4 = 20.
So 5x = 20 when x = 4.
Check: 5x is a multiplication, not 54. ✓
Watch out
4x means 4 × x — it is not the number forty-something.
Aim: Substitute into expressions like 3x + 2 in the right order.
When an expression has two parts, like 3x + 2, do the multiplication first, then the addition or subtraction (that is the BIDMAS order). For 3x + 2 with x = 4: first 3 × 4 = 12, then 12 + 2 = 14.
For 2x + 3, multiply first then add: 5 × 2 = 10, then + 3 = 13.
Worked example
Substitute x = 5 into 2x + 3.
2x + 3 means (2 × x) + 3.
Replace x with 5: (2 × 5) + 3.
Multiply first: 2 × 5 = 10.
Then add: 10 + 3 = 13.
So 2x + 3 = 13 when x = 5. Check: multiply before you add. ✓
Watch out
Do the × before the + or −. 2x + 3 with x = 5 is 13, not 25.
Aim: Substitute where there is a power, or a fraction of the letter.
A power means multiply the number by itself: x2 means x × x, so if x = 3 then x2 = 9. A fraction in front means a fraction of the letter: ½x means half of x (the same as x ÷ 2), and ⅓x means a third of x. Do the power or the fraction first, then the rest.
Aim: Put the given value into a rule and find the result.
A formula is a rule with an equals sign, like P = 4L. To use it, substitute the value you are given and work out the other side. Formulas turn a real situation — a perimeter, a price, a conversion — into one quick calculation.
The formula P = 4L is a machine: feed in L = 6, multiply by 4, out comes P = 24.
Worked example
A square has perimeter P = 4L. Find P when the side L = 6 cm.
The formula is P = 4L, which means P = 4 × L.
Replace L with 6: P = 4 × 6.
Work it out: 4 × 6 = 24.
So P = 24 cm.
Check: a square with side 6 has four sides of 6, and 4 × 6 = 24. ✓
Watch out
Substitute into the side that has the letter, then work out the subject (here P).
Try one
The cost of n tickets is C = 9n pounds. Find C when n = 7.
Aim: Substitute into formulas with two steps, including negative values.
Many formulas have more than one step — a multiply (or divide) and then an add. Work through them in order. The same method handles a negative value: replace the letter, keeping its sign, and follow the order of operations.
F = 9C ÷ 5 + 32 is a three-step machine: 20 × 9 = 180, ÷ 5 = 36, + 32 = 68.
Worked example
To change Celsius to Fahrenheit, F = 9C ÷ 5 + 32. Find F when C = 25.
Replace C with 25: F = 9 × 25 ÷ 5 + 32.
Work the multiply and divide first: 9 × 25 = 225, then 225 ÷ 5 = 45.
Then add: 45 + 32 = 77.
So F = 77, meaning 25°C = 77°F.
Check: do the × and ÷ before the + 32. ✓
Watch out
Keep the order: multiply and divide before adding. Carry a minus sign with its number.
Substitute negative values and into harder formulas.
Success criteria — I can:
carry the minus sign
follow the order of operations
do powers first
31.Substitute x = −2 into 3x + 10.
Answer:
Show working
Replace x with −2: 3 × (−2) + 10.
Multiply first: −6, then add 10 = 4.
So 3x + 10 = 4. ✓
32.Substitute x = −5 into 2x + 20.
Answer:
Show working
Replace x with −5: 2 × (−5) + 20.
Multiply first: −10, then add 20 = 10.
So 2x + 20 = 10. ✓
33.The formula v = u + at gives final speed. Find v when u = 4, a = 3 and t = 5.
Answer:
Show working
v = u + at. Replace: 4 + 3 × 5.
Multiply first: 3 × 5 = 15, then add 4 = 19.
So v = 19. ✓
34.Substitute x = −3 into x2 + 1.
Answer:
Show working
Replace x with −3: (−3)2 + 1.
A negative squared is positive: (−3) × (−3) = 9.
Then add 1: 9 + 1 = 10. So 10. ✓
35.A sequence is given by T = n2 + n. Find the 6th term (n = 6).
Answer:
Show working
T = n2 + n. Replace n with 6: 62 + 6.
62 = 36, then add 6 = 42.
So the 6th term is 42. ✓
0 of 35 answered
Guide
What is substitution in maths?
Substitution in maths means replacing a letter in an expression or formula with a number, then working out the value. You substitute into an expression — a set of terms with no equals sign, like 3x + 2 — and into a formula — a rule with an equals sign, like P = 4L for the perimeter of a square. Whatever the letter, you put the given number in its place and calculate.
The order matters. Work out any powers or brackets first, then multiplying and dividing, then adding and subtracting — the usual BIDMAS order. So for 3x + 2 with x = 4, you multiply 3 × 4 = 12 before adding 2 to reach 14. A term like 4x always means 4 × x, and x2 means x × x.
Common mistakes are reading 4x as a single number instead of a multiplication, adding before multiplying, squaring by doubling (treating x2 as x × 2), and dropping a minus sign when the value is negative. At Improve Tuition, a qualified teacher shows a pupil exactly how to replace each letter and follow the order of operations, in small worked steps — with read-aloud, comfort spacing and a reading tint for pupils who take in maths more easily that way.
Common questionsWhat is the difference between an expression and a formula?
An expression is a collection of terms with no equals sign, such as 3x + 2. A formula is a rule that does have an equals sign, such as P = 4L. You substitute into both the same way — by replacing the letter with a number.
How do you substitute a number into an expression?
Replace every letter with its value, keeping everything else the same, then work the calculation out in the usual order. For 2x + 3 with x = 5, that is 2 × 5 + 3 = 13.
What does x2 mean when you substitute?
x2 means x × x, not x × 2. So if x = 3, then x2 = 3 × 3 = 9. Always work out the power before multiplying or adding.
Which operation do you do first when substituting?
Follow BIDMAS: powers and brackets first, then multiply and divide, then add and subtract. In 3x + 2 you multiply 3 by the value before adding 2.
How do you substitute a negative number?
Put the negative value in place of the letter, keeping its sign, then follow the order of operations. For 3x + 10 with x = −2, that is 3 × (−2) + 10 = −6 + 10 = 4.
Stuck on this topic?
A teacher can find the exact gap
Practising substitution on your own is a strong start. If the same marks keep slipping, a qualified teacher can pinpoint the precise gap and fix it. Improve Tuition offers one-to-one maths tuition with our maths tutors in Batley and online — the first assessment is free.