Translate situations into algebra and substitute values
Represent quantities with letters, write expressions and formulas from words, and substitute signed values with brackets and units.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which expression means ‘three more than twice x’?
What you will learn
- Use letters for changing or unknown quantities
- Translate simple statements into expressions and formulas
- Substitute signed and fractional values accurately
Read the structure before writing symbols
A term is separated by plus or minus signs; 4n means 4 × n. ‘Five more than twice x’ is 2x + 5, while ‘five times x + 2’ is 5(x + 2). Brackets show what belongs together.
A formula relates quantities. Write it before substituting, keep units consistent, and distinguish a variable from a fixed coefficient.
Substitute with brackets
Replace every occurrence of a variable. If x = −3, then x² = (−3)² = 9; apply the normal order of operations after substitution.
Extended candidates should interpret more involved expressions and formulas. Always attach units to contextual results.
Pause and explain
If p = −4, what is p²?
Worked example
A fare is F = 4 + 1.8d dollars. Find F for d = 7.5 km and evaluate 2x² − 3x for x = −2.
Show the worked solution
- F = 4 + 1.8(7.5) = 17.5.
- For the second expression, 2(−2)² − 3(−2).
- Evaluate powers and products: 8 + 6 = 14.
Answer Fare $17.50; expression value 14.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A rectangle has length x + 4 and width x. Find its perimeter and area for x = 6 cm.
Give me a hint
Write formulas first, then substitute and keep linear and square units distinct.
Compare my reasoning
- P = 2((x + 4) + x) = 4x + 8.
- At x = 6, P = 32 cm.
- A = x(x + 4) = 6(10) = 60 cm².
Perimeter 32 cm; area 60 cm².
Look for these in your work
- Both dimensions used
- Perimeter formula correct
- Area has square units
Evaluate 3m² − 2n and 2(m − n) for m = −3, n = 5.
Give me a hint
Use brackets around the negative value and preserve the order in the second expression.
Compare my reasoning
- 3(−3)² − 2(5) = 27 − 10 = 17.
- 2(m − n) = 2(−3 − 5).
- 2(−8) = −16.
17 and −16.
Look for these in your work
- Negative square evaluated correctly
- Subtraction order preserved
- Whole bracket multiplied
A service charges $6 plus $1.25 per film. Write a formula, find the cost of 8 films, and find how many films cost $21.
Give me a hint
Let n be the film count; use the formula for both substitution and solving.
Compare my reasoning
- C = 6 + 1.25n.
- For n = 8, C = $16.
- 21 = 6 + 1.25n gives n = 12.
C = 6 + 1.25n; 8 films cost $16; 12 films cost $21.
Look for these in your work
- Fixed and variable charges separated
- Formula substituted correctly
- Equation solution interpreted as a count
Common mistakes
- Reading 5(x + 2) as 5x + 2
- Substituting a negative without brackets
- Combining unlike terms
- Dropping contextual units
What can you explain now?
Evaluate 3a − 2b for a = −1 and b = 4.
Compare with the explanation
−11
3(−1) − 2(4) = −3 − 8.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do brackets help with negative substitution, and how do words determine the order of operations?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.