Formula and Equation Synthesis
Choose the right algebra strategy for mixed practical problems: substitute, solve, rearrange, build a formula or test a model.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
A question gives a formula, a table, a total cost and several numbers. What do you do first so you do not use the wrong strategy?
Before calculating — write a decision process.
Before calculating, decide what the question is asking for. Different algebra problems need different first moves.
Need an output? Substitute values into the formula.
Need an unknown input? Write and solve an equation.
Wrong subject? Rearrange first, then substitute.
Need a formula from data? Find starting value and rate.
Key facts
- Different algebra problems need different first moves.
- Units, variable definitions and context guide the strategy.
- A final answer should be checked for reasonableness.
Concepts
- Substitution is used when the formula subject is already the required value.
- Rearranging is useful when the required variable is not the subject.
- Equations are needed when a total is known and an unknown input must be found.
Skills
- Choose between substituting, solving, rearranging and building formulas.
- Solve mixed practical algebra problems.
- Explain whether an answer is reasonable in context.
Before calculating, decide what the question is asking for. The table below summarises the key strategies:
| Question asks for… | Best first move | Example |
|---|---|---|
| A total from known inputs | Substitute | $C = 12 + 4r$, find $C$ when $r = 7$ |
| The input that produced a total | Solve an equation | $12 + 4r = 40$ |
| A variable not currently alone | Rearrange | $d = st$, find $s$ |
| A formula from data | Find starting value and rate | Table outputs increase by 5 |
What to write in your book
- Strategy guide: Need output? Substitute. Need unknown input? Solve. Wrong subject? Rearrange. Need formula from table? Find start value and rate.
- Always state the strategy before calculating.
- Define your variable with its units (e.g. "Let $p$ = number of pages").
- Check reasonableness: does the answer make sense in context?
Quick check: A total cost is known and the number of items is unknown. Which strategy should you use first?
After obtaining an answer, always ask: does this make sense?
A stopping-distance model is $D = 0.01v^2 + 0.3v$. If a student says the stopping distance at 60 km/h is 540 m, check: $D = 0.01(3600) + 18 = 36 + 18 = 54$ m. The student's answer is ten times too large.
What to write in your book
- After every calculation: check units, check the magnitude, and check the sign.
- A ten-times error often means a misplaced decimal point.
- For worded problems: re-read the question and confirm the answer answers what was asked.
True or false: If an answer seems unreasonably large, the error is always a sign mistake.
Worked examples · 4 in a row, reveal as you go
A printer charges $25 setup plus $2 per page. A job costs $81. How many pages were printed?
A cyclist travels 135 km in 3 hours. Use $d = st$ to find the average speed.
| Hours, $h$ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Cost, $C$ | $30 | $42 | $54 | $66 |
A stopping-distance model is $D = 0.01v^2 + 0.3v$. A student says the stopping distance at 60 km/h is 540 m. Is this correct?
What to write in your book
- WE1: 28 pages. Strategy: write and solve $25 + 2p = 81$.
- WE2: 45 km/h. Strategy: rearrange $d = st$ to $s = d \div t$ first.
- WE3: $C = 30 + 12h$. Start = 30, rate = 12.
- WE4: Correct answer is 54 m, not 540 m. Decimal error — always check reasonableness.
Fill the gap: A table has outputs 8, 13, 18, 23 for inputs 0, 1, 2, 3. The starting value is and the rate of change is , so the formula is $y =$ .
Common errors · the 3 traps that cost marks
Quick-fire practice · 4 calculations
A gym charges $20 plus $15 per class. Find the number of classes if the total is $110.
Use $A = s^2$ to find the area of a square with side length 9.5 m.
Rearrange $A = bh$ to find $h$ when $A = 72$ cm² and $b = 9$ cm.
Write a formula for outputs 8, 13, 18, 23 for inputs 0, 1, 2, 3.
Odd one out: Three of these statements about the strategy guide are correct. Which one is wrong?
In your own words: Describe the correct first move when you are given a formula and need to find the value of a variable that is not the subject.
A reliable first move is: identify the required value, define the variable if needed, choose the strategy, calculate, then interpret the answer with units.
Earlier you wrote a decision process before seeing the lesson. Compare it to what you now know.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Choose a strategy, show working and interpret the answer.
Q1. A hire company charges $35 plus $18 per hour. The total cost is $143. Find the hire time and explain your strategy. (4 marks)
Q2. Use $d = st$ to find time when $d = 210$ km and $s = 70$ km/h. Rearrange before substituting. (3 marks)
Q3. A table has outputs 14, 20, 26, 32 for inputs 0, 1, 2, 3. Write a formula and test it using input 3. (4 marks)
📖 Answers (click to reveal)
Drill 1: Strategy: solve an equation. $20 + 15c = 110$, $15c = 90$, $c = 6$ classes.
Drill 2: $A = (9.5)^2 = 90.25$ m².
Drill 3: Rearrange $A = bh$ to $h = A \div b = 72 \div 9 = 8$ cm.
Drill 4: Start = 8, rate = 5. Formula: $y = 8 + 5x$ (or equivalent).
Q1 (4 marks): Strategy: write and solve an equation [1]. Let $h$ = hire time in hours. $35 + 18h = 143$ [1]. $18h = 108$, $h = 6$ [1]. The hire time is 6 hours [1].
Q2 (3 marks): Rearrange $d = st$ to $t = d \div s$ [1]. Substitute: $t = 210 \div 70$ [1]. $t = 3$ hours [1].
Q3 (4 marks): Start = 14 [1]. Rate = $20 - 14 = 6$ per input [1]. Formula: $y = 14 + 6x$ [1]. Test: $y = 14 + 6(3) = 32$ ✓ [1].
Sort each question into substitute, solve, rearrange or build formula before doing any calculation. Beat the boss to bank a tier.
⚔ Enter the arenaClimb platforms by answering mixed algebra strategy questions. Pool: lesson 8.
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