An expression is a mathematical phrase built from numbers, letters (variables) and operations. Unlike an equation, it has no equals sign — it just represents a value that can change.
No equals signVariables + numbersExpressions vs equationsReal engineering context
Each expression represents a quantity whose value depends on what the variable(s) equal
Expression vs EquationThe crucial difference
This is the most important distinction to get right from the start:
EXPRESSION: 3x + 5 (no = sign)
EQUATION: 3x + 5 = 17 (has = sign)
FORMULA: A = l × w (named equation)
✅ An expression is like a recipe — it tells you what to do with a value. An equation says two things are equal, so you can solve it to find the unknown.
⚠️ You can simplify an expression. You can solve an equation. Don’t mix up the two words!
Why do engineers use expressions?
Generalise — instead of saying “force = 3 × 4 = 12N”, write F = ma to cover every possible case
Communicate — V = IR works for every voltage, current and resistance combination ever
Calculate efficiently — write the expression once, substitute different values as needed
Design — “what value of r makes A = 50?” requires knowing the expression A = πr²
📌 In T Level / Level 3 Engineering you’ll use expressions in stress (σ = F/A), gear ratio, electrical power and beam deflection calculations.
Real Engineering ExpressionOhm’s Law — V = IR is an expression for voltage
💡 The expression V = 2R tells us everything — it works for any value of R. We don’t need to write a new calculation every time!
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Anatomy of a Term
A term is a single chunk in an expression — separated from other terms by + or −. Every term has up to three parts: a coefficient, a variable and a power.
The Three PartsBreaking down the term 5x² completely
5x² means “5 multiplied by x multiplied by x” — i.e. 5 × x × x
If x = 3: 5 × 3 × 3 = 5 × 9 = 45
CoefficientThe number in front of the variable
5x² → coefficient = 5
−3y → coefficient = −3
x → coefficient = 1 (invisible!)
−x → coefficient = −1 (invisible!)
💡 If there is no number written in front of x, the coefficient is 1. So x = 1x and −x = −1x. This trips lots of people up!
⚠️ The coefficient is always the number directly attached to the variable. In 3 + x, the 3 is a separate constant term — the coefficient of x is still 1!
Power / IndexThe small raised number (also called an exponent)
x³ → power = 3 → x × x × x
x² → power = 2 → x × x
x → power = 1 (invisible!)
x⁰ → power = 0 = 1 always
💡 If there is no power written, the power is 1. So x = x¹. You’ll need this when applying the indices laws!
📌 The power is called an index (plural: indices) in UK maths. It’s also called an exponent in some textbooks.
Practice IdentifyingIdentify the coefficient, variable and power in each term
✅ Notice that ab means a × b — two variables multiplied together. The coefficient is 1 (invisible), both powers are 1 (invisible).
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Like & Unlike Terms
Like terms have exactly the same variable AND the same power. Only like terms can be added or subtracted together. Think of it like units — you can add metres to metres, but not metres to seconds!
The RuleTwo terms are LIKE if they match on both counts
Like terms must have: same variable(s) AND same power
Both conditions must be true — matching one is not enough!
3x and 7x ✔ same variable (x), same power (1)
4x² and 9x² ✔ same variable (x), same power (2)
5ab and 3ab ✔ same variables (a and b), same powers
❌ 3x and 3x² — same variable but DIFFERENT powers → NOT like
❌ 4x and 4y — different variables → NOT like
❌ 5x² and 5x³ — different powers → NOT like
AnalogyThink of terms like physical units
You can add same units together, but you cannot mix different units:
3 metres + 5 metres = 8 metres ✔
4 kg + 7 kg = 11 kg ✔
3 metres + 4 kg = ??? ❌ different units!
📌 In exactly the same way:
3x + 5x = 8x ✔ (both “x units”)
3x + 5y = 3x + 5y ❌ cannot combine (different)
💡 This analogy is exactly why engineers must keep units consistent in calculations — it’s the same mathematical principle!
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Collecting Like Terms
This is the main simplification skill in algebra. You group and combine like terms to write an expression in its simplest form. The expression’s value never changes — it just looks tidier.
5x − 2x = 3x | 3y + 7y = 10y | constant: −4
∴ 3x + 10y − 4 ✅
Step-by-Step MethodAlways follow this order
Identify all the different types of term (x terms, y terms, x² terms, constants…)
Underline each group in a different colour if it helps
Add or subtract the coefficients within each group
Write each simplified group in your answer
Check you haven’t accidentally dropped a term
✅ You don’t need to rewrite in a specific order, but convention is: highest power first, then descending, then constants at the end.
Common MistakesThe errors that cost marks
❌ Combining unlike terms: 3x + 4y = 7xy → WRONG! 3x and 4y are unlike → leave them separate
❌ Forgetting the sign: 5x − 2x = 3x ✔ but students write 5x − 2x = 7x ❌ The minus belongs to the 2x term!
❌ Mixing x and x²: 3x + 4x² = 7x² → WRONG! They have different powers → cannot combine
✅ When in doubt: write out the coefficients with their signs explicitly before adding.
Worked Example 2 — Simplify: 4a² + 3b − a² + 2b − 5
⚙️ Engineering Example — Total resistance: R₁ + R₂ + R₁ − R₂ + 3R₃
Context: Simplify an expression for total resistance in a circuit
R₁ terms: R₁ + R₁ = 2R₁
R₂ terms: R₂ − R₂ = 0 (they cancel!)
R₃ terms: 3R₃
Answer:RTOTAL = 2R₁ + 3R₃ ✅
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Substitution
Substitution means replacing the variable(s) with actual numbers to find the value of the expression. This is what engineers do every time they “plug numbers into a formula”.
Try it yourselfLive substitution calculator
ℹ️ How to use: Type any number into the x = box. The expression 3x² − 2x + 1 is evaluated instantly with full step-by-step working shown below.
EXPRESSION
3x² − 2x + 1
x =
👆Type any number into the box — the steps and answer update instantly. Try −2 to see negative substitution in brackets · try 0.5 for a decimal · try 0 to see the constant term left on its own.
MethodHow to substitute safely — always use brackets!
Write out the expression
Replace every instance of the variable with the value in brackets
Work out each term using BODMAS
Combine to get the final value
Find 3x² − 2x + 1 when x = 4
= 3(4)² − 2(4) + 1
= 3(16) − 8 + 1
= 48 − 8 + 1 = 41
💡 Always put the substituted value in ( ) brackets. This prevents sign errors, especially with negative values and powers.
Negative value substitutionThe trickiest type — brackets save you!
⚠️ When substituting a negative number, the brackets are essential: x = −3 in 2x²: 2(−3)² = 2 × 9 = 18 ✔ Without brackets: 2 × −3² might be read as −18 ❌
Invisible coefficients: x = 1x, −x = −1x
Invisible powers: x = x¹, treat as power 1
Quick ReferenceCan you combine these? — instant decision guide
🎯 Exam tip: In T Level papers, expressions questions often ask you to “simplify” or “write in simplest form”. That always means collect like terms. If you see “evaluate” or “find the value”, that means substitute the given number in.
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Quick Fire Quiz
Test your knowledge — 10 questions, instant feedback.
⚡ Expressions Blitz
ℹ️ How to play: A question appears on screen. Click the answer you think is correct from the four options. Your score and streak update after each question. Click Next to move on.
Score: 0Streak: 0 🔥Q 1/10
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👆Click an answer button to lock it in — correct answers turn green, wrong ones turn red and reveal the right answer · Build a streak 🔥 by answering correctly in a row · Press Play Again after question 10 to reshuffle and try for a higher score.
Score: 0/10
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Applications
Where this topic is used in engineering, manufacturing, maintenance and daily life.
⚡Electrical:V=IR — write and evaluate for I=2, R=5.
🌡️Maintenance:Repair cost C=35h+80. Evaluate for h=3 hours.
🛒Daily Life:Mobile bill C=0.05m+12. How much for 200 minutes?
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Practice Questions
15 questions across 3 tiers. Click to reveal the full working. Aim to attempt each one before looking at the answer!
🟢 Tier 1 — Foundations
ℹ️ How to use: Work through each question and write down your answer. When you are ready, click Show Answer to reveal the full worked solution and mark scheme.
Q1
Write down the coefficient and power of x in the term: −7x³
[2]
Coefficient = −7 Power of x = 3 (The minus sign belongs to the coefficient)
Q2
Write down the coefficient of x in: x − 4
[1]
Hint: what number is invisibly in front of x?
x means 1x, so coefficient = 1
Q3
Which of these pairs are like terms?
(a) 4x and −9x (b) 3x² and 3x (c) 5ab and 2ab (d) 7y and 7z
[4]
(a) ✔ LIKE — both x, both power 1
(b) ❌ UNLIKE — x² and x have different powers
(c) ✔ LIKE — both have variables a and b, same powers
(d) ❌ UNLIKE — different variables (y vs z)
Q4
Simplify: 8x + 3x
[1]
Both x terms: 8 + 3 = 11 Answer: 11x
Q5
Evaluate 5x − 3 when x = 4
[2]
5(4) − 3 = 20 − 3 = 17
🟡 Tier 2 — Core Skills
Q6
Simplify: 6x + 4y − 2x + y − 3
[3]
x terms: 6x − 2x = 4x y terms: 4y + y = 5y Constant: −3 Answer: 4x + 5y − 3
A student writes: “4x + 3y = 7xy”. Explain the error and write the correct simplified form.
[2]
Error: 4x and 3y are UNLIKE terms (different variables). They cannot be combined — the expression is already in simplest form. Correct answer: 4x + 3y (cannot be simplified)
Using stress formula σ = F/A, find the stress when F = 3x + 7 kN and A = 2 mm². Write the stress as an expression, then find the value when x = 4.
[4]
σ = F/A = (3x + 7)/2 (expression for stress in kN/mm²) When x = 4: σ = (3(4) + 7)/2 = (12 + 7)/2 = 19/2 = 9.5 kN/mm² = 9.5 GPa ✅
Q13
The perimeter of a rectangle is 2(3x + 5) + 2(x − 2). Expand and simplify. Find the perimeter when x = 6.
[4]
2(3x+5) = 6x + 10 2(x−2) = 2x − 4 Total = 6x + 10 + 2x − 4 = 8x + 6 When x = 6: 8(6) + 6 = 48 + 6 = 54 units ✅
Q14
Using KE = ½mv², write an expression for KE when m = 2x kg and v = 3x m/s. Simplify fully.
[3]
KE = ½ × (2x) × (3x)² = ½ × 2x × 9x² = ½ × 18x³ = 9x³ J
Q15
A CNC machine cuts metal at a rate given by the expression: R = 4v² − 3v + 2 mm/min where v is the spindle speed in hundreds of rpm. Find R when v = 5, and explain what the answer means in context.
[4]
R = 4(5)² − 3(5) + 2 = 4(25) − 15 + 2 = 100 − 15 + 2 = 87 mm/min Meaning: at 500 rpm (v=5), the machine removes material at 87 mm per minute. ✅