SECONDARY 1 MATHEMATICS LEARNING GUIDE · GUIDE 15
A formula is a compact statement of a relationship. It tells how one quantity depends on others. Substitution replaces variables with known values. Rearrangement changes which variable is isolated while preserving the relationship.
These skills become increasingly important because Secondary Mathematics uses symbols to compress meaning. A student who treats formulae as disconnected recipes may calculate correctly in familiar questions but become stuck when the subject changes, a negative value is substituted or the formula must be rearranged.
This guide connects directly to Algebraic Expressions and Variables and Equations and Equality. Return to the Secondary Mathematics Hub for the full route.
Navigate: formula meaning · substitution · negative values · units · rearrangement · multi-step rearrangement · practice · answers.
1. Variables have roles inside a formula
In A = lw, A represents area, l length and w width. The letters are not arbitrary marks once the formula has been defined.
A formula can be read as a sentence: “area equals length multiplied by width”. This verbal reading helps prevent operations from being forgotten.
Worked contrast
P = 2l + 2w gives a rectangle’s perimeter. A = lw gives its area. The same variables appear, but the relationships are different because the quantities being measured are different.
2. A formula can produce many valid numerical cases
The formula C = 3n + 5 describes a family of values. If n = 0, C = 5. If n = 4, C = 17. If n changes, C changes according to the same rule.
This is different from an equation such as 3n + 5 = 17, which asks which n makes the relationship equal to 17.
Formula versus equation-to-solve
A formula often defines a dependent quantity in terms of others. A specific equation may then be formed from it when one value is known and another is unknown.
3. Substitution means replacing a variable with its complete value
If y = 4x − 7 and x = 5, then y = 4(5) − 7 = 20 − 7 = 13.
Write the substituted value in brackets when that helps preserve signs or structure.
Worked example with two variables
Evaluate P = 2a + 3b when a = 4 and b = 7.
P = 2(4) + 3(7) = 8 + 21 = 29.
4. Negative values need brackets
Suppose f = x² + 3x and x = −4.
f = (−4)² + 3(−4) = 16 − 12 = 4.
Writing −4² without brackets would normally mean −(4²), which is −16. The brackets record that the complete value −4 replaces x.
Another example
Evaluate 2a − b² for a = −3 and b = −2:
2(−3) − (−2)² = −6 − 4 = −10.
5. Fractions and decimals can be substituted exactly
Evaluate y = 3x + 1 when x = 2/3.
y = 3(2/3) + 1 = 2 + 1 = 3.
Keeping the fraction exact makes cancellation clear.
Decimal example
If d = 5t and t = 1.2, then d = 5(1.2) = 6.
6. Formulae carry units as well as numbers
If d = vt, distance equals speed multiplied by time. A speed of 12 km/h used for 0.5 h gives d = 6 km.
If time were given in minutes, it would need conversion before combining it with km/h.
Unit check
km/h × h = km. The time units cancel, leaving a distance unit.
This kind of dimensional check can reveal a wrong operation before the final answer.
7. Formulae can encode geometry
For a triangle, A = 1/2 bh. If b = 10 cm and h = 7 cm, then A = 35 cm².
The formula depends on h being perpendicular to the chosen base. Substitution cannot repair a wrong geometric interpretation.
Meaning before arithmetic
Every variable should be matched to the correct measured quantity before values are inserted.
8. Rearrangement preserves equality
Suppose y = x + 6 and we want x as the subject. Subtract 6 from both sides:
x = y − 6.
The relationship has not changed. We have simply written it in a form that makes x easy to calculate.
Worked example: multiplication
If A = bh and h is required, divide both sides by b, assuming b ≠ 0:
h = A/b.
9. Undo operations in a controlled order
Consider y = 3x + 5. To make x the subject, first subtract 5, then divide by 3:
y − 5 = 3x
x = (y − 5)/3.
Why the bracket matters
The entire quantity y − 5 is divided by 3. Writing y − 5/3 would mean only 5 is divided by 3.
10. Rearranging a formula with subtraction
Given P = 2l + 2w, make l the subject.
Subtract 2w: P − 2w = 2l.
Divide by 2: l = (P − 2w)/2.
This can also be written l = P/2 − w. Both forms are equivalent.
11. Brackets can be treated as a grouped object
Given y = a(x + b), make x the subject.
Divide by a, assuming a ≠ 0:
y/a = x + b.
Subtract b:
x = y/a − b.
Alternative form
x = (y − ab)/a is algebraically equivalent when a ≠ 0, but the first route mirrors the original structure more transparently.
12. Fractions in formulae require careful clearing
Given A = 1/2 bh, make h the subject.
Multiply both sides by 2: 2A = bh.
Divide by b: h = 2A/b, with b ≠ 0.
Check by substitution
If A = 30 and b = 12, then h = 60/12 = 5. Forward check: 1/2 × 12 × 5 = 30.
13. Formulae with a denominator can be rearranged by multiplication
Given v = d/t, make d the subject. Multiply by t:
d = vt.
Make t the subject instead:
vt = d, so t = d/v, assuming v ≠ 0.
The familiar speed formulas are therefore rearrangements of one relationship.
14. A variable can appear more than once
Consider P = 2l + 2w. The subject l appears once, so isolation is direct. But in A = x(y + z), x still appears once even though other variables are grouped.
More advanced formulas may contain the target variable in several places, requiring collection or factorisation. Treat those as later extensions unless your course has introduced them.
15. Check a rearrangement by numerical substitution
Suppose C = 4n + 7 is rearranged to n = (C − 7)/4.
Choose n = 5. Then C = 27. Substitute C = 27 into the rearranged form: n = (27 − 7)/4 = 5.
A numerical check cannot prove a symbolic rearrangement in full generality, but it can catch many sign and bracket errors.
16. Check symbolically by substituting back
If x = (y − 5)/3 came from y = 3x + 5, substitute the new expression for x into the original:
3[(y − 5)/3] + 5 = y − 5 + 5 = y.
The original relationship is recovered.
17. Common formula errors
| Error | Likely issue | Repair prompt |
|---|---|---|
| Substitutes x = −3 as −3² | Complete negative value not bracketed | What exactly replaces x? |
| Writes x = y − 5/3 from y = 3x + 5 | Whole numerator not grouped | What quantity is divided by 3? |
| Changes side and sign without preserving equality | Transposition slogan replacing operations | What operation is applied to both sides? |
| Uses minutes with km/h | Units incompatible | What unit does the formula require? |
| Gets the subject but forgets requested evaluation | Intermediate result treated as final | What does the question actually ask for? |
18. Practice laboratory
- Evaluate y = 3x + 5 for x = 4.
- Evaluate y = x² − 2x for x = −3.
- Evaluate P = 2a + 5b for a = −2 and b = 3.
- Evaluate A = 1/2 bh for b = 12 and h = 7.
- Make x the subject of y = x + 9.
- Make x the subject of y = 5x.
- Make x the subject of y = 4x − 3.
- Make h the subject of A = bh.
- Make w the subject of P = 2l + 2w.
- Make h the subject of A = 1/2 bh.
- Make t the subject of d = vt.
- Make r the subject of C = 2πr.
- Given C = 5n + 8, find n when C = 43.
- Given A = 1/2 bh, find b when A = 54 and h = 9.
- Check whether x = (y + 2)/6 is a correct rearrangement of y = 6x − 2.
19. Explained answers
1. 3(4) + 5 = 17.
2. (−3)² − 2(−3) = 9 + 6 = 15.
3. 2(−2) + 5(3) = −4 + 15 = 11.
4. 1/2 × 12 × 7 = 42.
5. x = y − 9.
6. x = y/5.
7. y + 3 = 4x, so x = (y + 3)/4.
8. h = A/b, b ≠ 0.
9. P − 2l = 2w, so w = (P − 2l)/2.
10. h = 2A/b, b ≠ 0.
11. t = d/v, v ≠ 0.
12. r = C/(2π).
13. 43 = 5n + 8, so n = 7.
14. 54 = 1/2 × b × 9. Then 108 = 9b, so b = 12.
15. Yes. From y = 6x − 2, add 2 and divide by 6 to obtain x = (y + 2)/6.
20. Complete mixed problem
Problem: A fictional rectangular frame has perimeter P = 2l + 2w. The perimeter is 54 cm and width is 9 cm. Find the length by rearranging first, then substituting.
Rearrange:
P − 2w = 2l, so l = (P − 2w)/2.
Substitute P = 54 and w = 9:
l = (54 − 18)/2 = 36/2 = 18 cm.
Check: 2(18) + 2(9) = 36 + 18 = 54 cm.
21. Teaching formulae as reversible relationships
Use one formula in several directions. With d = vt, first find d, then find v, then find t using different known values.
This prevents the learner from storing three disconnected formulas and instead builds the idea that one relationship can be rearranged.
Use units as a check
Ask what unit should remain after the operation. This makes formula selection inspectable.
22. Questions students often ask
Is rearranging the same as moving a term across?
Informally it may look that way, but the reliable reasoning is to apply inverse operations while preserving equality.
Why do I need brackets around a negative substitution?
Because the entire signed value replaces the variable and may be affected by powers or multiplication.
Can two rearranged formulae look different but both be correct?
Yes, if they are algebraically equivalent.
What if the denominator becomes zero?
Then that rearranged form is not defined for that value. State the non-zero condition when relevant.
What is the best check?
Substitute simple values into both the original and rearranged formula, or substitute the rearranged expression back symbolically.
23. Return path
Formulae bring together variables, equality, units and modelling. Revisit Algebraic Expressions and Variables for symbolic meaning, Equations and Equality for transformation logic and Mensuration for geometric formula applications.
Sources and learning boundaries
Official curriculum reference: MOE Secondary Syllabus Directory. Topic sequencing and the complexity of changing the subject vary across subject levels and schools.
The worked examples and applications are independently constructed teaching examples.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Define each variable, preserve the relationship, substitute complete values, verify units and reverse the formula when the unknown changes.
