Algebraic language and substitution
Use letters to express a relationship, then replace them with values without changing the structure.
Core and Extended.
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
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources