Algebraic language and substitution
Use letters to express a relationship, then replace them with values without changing the structure.
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Apply brackets and the order of operations.
- Multiply and square negative numbers.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is (−3)²?
What you will learn
- Distinguish expressions, equations and formulas.
- Translate a description into an expression.
- Substitute signed values accurately.
A letter can vary
In 4n + 3, n stands for a number and 4n means four multiplied by that number. The expression gives a rule that works for many values of n. The coefficient is 4 and the constant term is 3. The terms are separated by addition or subtraction, so 4n + 3 has two terms.
An expression has no equality to solve. An equation such as 4n + 3 = 19 states that two values are equal and asks which n makes that statement true. A formula, such as A = lw, expresses a general relationship between named quantities. State what each letter means and its units when modelling a situation.
Translate the order, not just the words
Five less than twice x is 2x − 5; twice five less than x is 2(x − 5). Brackets change what is doubled. For a number n, the next consecutive integer is n + 1. If an even number is 2n, the next even number is 2n + 2, rather than 2n + 1.
Multiplication is often left implicit: ab means a × b, but a + b is a different operation. A fraction bar groups a numerator and denominator. Read (x + 2)/3 as dividing the whole sum by three. Writing the relationship in words before symbols helps expose whether a fee, discount or change acts on one part or on the whole.
Replace each letter with a bracketed value
To evaluate 2x² − 3x at x = −2, write 2(−2)² − 3(−2). The square is 4, so the result is 8 + 6 = 14. Replacing x by −2 without brackets can wrongly turn its square into −4. If a letter appears twice, replace both occurrences with the same stated value.
For A = (a + b)h/2, the sum is formed before multiplying by h and dividing by two. Insert units only after the arithmetic structure is clear. Substitution checks whether an equation is satisfied: for 3x + 1 = 10 and x = 3, both sides are ten. It does not prove a formula works for every possible input.
Pause and explain
Which expression means three less than twice n?
Worked example
A taxi costs $4 plus $2.50 per kilometre. Write a cost formula and find the cost of 6 km.
Show the worked solution
- Let d be the distance in kilometres and C the cost in dollars.
- The fixed charge appears once: C = 4 + 2.5d.
- For d = 6, C = 4 + 2.5 × 6 = 19.
Answer C = 4 + 2.5d; $19
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.
Evaluate 3a − 2b when a = −4 and b = 5.
Give me a hint
Substitute with brackets before combining terms.
Compare my reasoning
- Write 3(−4) − 2(5).
- The two products are −12 and 10.
- Subtract: −12 − 10 = −22.
−22
Look for these in your work
- Substituted both values correctly.
- Kept the subtraction of a positive product.
Evaluate (p² + q)/3 when p = −5 and q = 2.
Give me a hint
The fraction bar groups the entire numerator.
Compare my reasoning
- The square (−5)² is 25.
- The numerator is 25 + 2 = 27.
- Divide the whole numerator by 3: 27/3 = 9.
9
Look for these in your work
- Squared the bracketed negative value.
- Divided the full numerator.
A hall has r rows of s seats and 8 separate chairs. Write its total capacity and find it for r = 12 and s = 15.
Give me a hint
The separate chairs are added once after counting the row seats.
Compare my reasoning
- The rows contain rs seats.
- Total capacity is N = rs + 8.
- N = 12 × 15 + 8 = 188 seats.
N = rs + 8; 188 seats
Look for these in your work
- Multiplied the two dimensions of the arrangement.
- Added the fixed extra chairs once.
Common mistakes
- Reading 4n as 4 + n.
- Omitting brackets around a negative substituted value.
- Trying to solve an expression that has no equation.
What can you explain now?
Evaluate 2t² + t for t = −3.
Compare with the explanation
15
2(−3)² + (−3) = 18 − 3 = 15.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, translate two similar sentences with different bracket placement. Substitute a negative value into both and explain why their results differ.
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.