Formula and Equation Synthesis

Choose the right algebra strategy for mixed practical problems: substitute, solve, rearrange, build a formula or test a model.

45 min Algebra Formulas and equations Lesson 8 of 13

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Use the printable version for mixed formula and equation questions with full working and reasonableness checks.

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Think First

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?

Type a decision process before calculating.

Write a decision process in your book before calculating.

Write your decision process in your book

Saved

Know

Different algebra problems need different first moves.
Units, variable definitions and context guide the strategy.
A final answer should be checked for reasonableness.

Understand

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.

Can Do

Choose between substituting, solving, rearranging and building formulas.
Solve mixed practical algebra problems.
Explain whether an answer is reasonable in context.

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Strategy Decision Guide

Need an output?

Substitute values into the formula.

Need an unknown input?

Write and solve an equation.

Wrong subject?

Rearrange first, then substitute.

Core Content

1. Choose the First Move

Before calculating, decide what the question is asking for.

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

Common error: Do not try to use every number immediately. Identify the required value first.

Worked Example 1

Mixed cost problem

A printer charges $25 setup plus $2 per page. A job costs $81. How many pages were printed?

Strategy: The total is known and the number of pages is unknown, so write an equation.

Let $p$ be the number of pages.

$25 + 2p = 81$

$2p = 56$, so $p = 28$.

Answer: 28 pages were printed.

Worked Example 2

Rearrange before substituting

A cyclist travels 135 km in 3 hours. Use $d = st$ to find the average speed.

Strategy: Speed is not the subject, so rearrange first.

$d = st$ becomes $s = \frac{d}{t}$.

$s = \frac{135}{3} = 45$.

Answer: The average speed is 45 km/h.

Worked Example 3

Build and test a formula from a table

Hours, h	0	1	2	3
Cost, C	$30	$42	$54	$66

Starting value: $30 when $h = 0$.

Repeated change: +$12 per hour.

Formula: $C = 30 + 12h$.

Test: if $h = 3$, $C = 30 + 12(3) = 66$, matching the table.

Worked Example 4

Check reasonableness

A stopping-distance model is $D = 0.01v^2 + 0.3v$. A student says the stopping distance at 60 km/h is 540 m.

$D = 0.01(60)^2 + 0.3(60)$

$D = 36 + 18 = 54$ m.

The student's answer is ten times too large. They likely multiplied by 0.1 or misplaced a decimal.

Reasonableness habit: Large differences often signal a decimal, unit or substitution error.

Activity

Mixed Strategy Practice

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.

Complete the mixed strategy practice in your book.

Revisit the Decision Process

A reliable first move is: identify the required value, define the variable if needed, choose the strategy, calculate, then interpret the answer with units.

Write your mixed-question checklist in your book.

Assessment

MC

Multiple Choice

Random questions from the lesson bank - feedback appears immediately.

SA

Short Answer

Choose a strategy, show working and interpret the answer.

1. A hire company charges $35 plus $18 per hour. The total cost is $143. Find the hire time and explain your strategy. 4 MARKS

Answer in your book.

2. Use $d = st$ to find time when $d = 210$ km and $s = 70$ km/h. Rearrange before substituting. 3 MARKS

Answer in your book.

3. 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

Answer in your book.

Strategy Sort

Sort each question into substitute, solve, rearrange or build formula before doing any calculation.

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