Practical simultaneous/inequality models and checks | Year 10

01 / LEARN

Understand the idea.

Build a model from the information

Define what each variable counts and its units before forming a model. A total count adds counts; a total cost adds price multiplied by count. These are different conditions, even when both involve the same variables.

Use equality for an exact requirement, $\leq$ for a maximum and $\geq$ for a minimum. State count restrictions as well: nonnegative whole numbers are required when variables represent complete tickets or manufactured items.

Interpret, not just calculate

After finding a candidate plan, check every original total or limit and the variable domain. A feasible plan satisfies all stated conditions, but need not be unique. Interpret the conclusion using the model assumptions; omitted fees or constraints could alter a real decision.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Ticket prices and attendance

An event sells 13 tickets: adult tickets cost 10 dollars and child tickets cost 6 dollars. The total is 98 dollars. How many of each were sold?

Worked solution

1
Define the counts and form both totals.
$\begin{aligned} a + c & = 13 \\ 10 a + 6 c & = 98 \end{aligned}$
Here $a$ counts adult tickets and $c$ counts child tickets; both must be nonnegative integers.
2
Substitute the count relation into the cost equation.
$10 a + 6 (13 - a) = 98$
The count total gives $c = 13 - a$ .
3
Solve for the adult count.
$\begin{aligned} 10 a + 78 - 6 a & = 98 \\ 4 a & = 20 \\ a & = 5 \end{aligned}$
Collecting terms separates the extra cost of adult tickets from the child-ticket baseline.
4
Find the child count.
$c = 13 - 5 = 8$
The two counts must sum to exactly $13$ .
5
Check both totals and the count domain.
$\begin{aligned} 5 + 8 & = 13 \\ 10 (5) + 6 (8) & = 50 + 48 = 98 \end{aligned}$
Five adults and eight children are nonnegative whole counts satisfying attendance and revenue.

Answer

5 adult tickets and 8 child tickets.

EXAMPLE 02

Check a proposed plan

A workshop can use at most 30 labour-hours. Item A takes 3 hours and item B takes 2 hours. At least 8 items are needed. Is making 6 of A and 4 of B feasible?

Worked solution

1
Write all constraints.
$\begin{aligned} 3 a + 2 b & \leq 30 \\ a + b & \geq 8 \end{aligned}$
Here $a$ and $b$ are nonnegative integer item counts; labour has a maximum and output has a minimum.
2
Check the labour requirement for the plan.
$3 (6) + 2 (4) = 26 \leq 30$
The proposed six A items and four B items fit within the $30$ available labour-hours.
3
Check the total output.
$6 + 4 = 10 \geq 8$
The plan meets the minimum number of items.
4
Check the count domain and conclude.
$a = 6, b = 4$
Both counts are nonnegative integers, so all stated conditions hold and the plan is feasible.

Answer

Yes: it uses 26 hours and produces 10 items.

03 / YOUR TURN

Try it yourself.

Check your understanding

Two notebooks and one pen cost 11 dollars. One notebook and one pen cost 7 dollars. Find each price.

Show answer & reasoning

Notebook: 4 dollars; pen: 3 dollars.

Let $n$ and $p$ be the notebook and pen prices in dollars. Subtract $n + p = 7$ from $2 n + p = 11$ to obtain $n = 4$ , then $p = 3$ . Check: $2 (4) + 3 = 11$ and $4 + 3 = 7$ .

Keep practising

6 worksheet questions · 20 marks

Open worksheet Worked solutions

Read the questions online

One smoothie and two juices cost 14 dollars. Two smoothies and two juices cost 20 dollars. Find both prices.
3 marks
A car park has 18 vehicles, all cars or motorbikes. There are 58 wheels. Find each number.
3 marks
A hire fee is 25 dollars plus 6 dollars per hour. Write the cost condition for a budget of at most 70 dollars, then find the greatest whole number of hours.
3 marks
A recipe uses 3 units of flour and 2 units of sugar per batch A, or 1 unit of flour and 3 units of sugar per batch B. Exact totals of 17 flour units and 16 sugar units are used. Find the batch counts.
4 marks
A model gives 7.5 buses for transporting a group. Explain why this is not a directly usable bus count and what extra information is needed.
3 marks
A council must buy at least 12 benches and tables in total. Benches cost 80 dollars and tables cost 120 dollars; the budget is 1200 dollars. Define variables, write constraints and test 9 benches with 3 tables.
4 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

One smoothie and two juices cost 14 dollars. Two smoothies and two juices cost 20 dollars. Find both prices.
3 marks
Subtract the totals to find a smoothie, then solve for two juices.
Smoothie: 6 dollars; juice: 4 dollars.
A car park has 18 vehicles, all cars or motorbikes. There are 58 wheels. Find each number.
3 marks
$c + m = 18$ and $4 c + 2 m = 58$ ; subtract twice the first to get $2 c = 22$ .
11 cars and 7 motorbikes.
A hire fee is 25 dollars plus 6 dollars per hour. Write the cost condition for a budget of at most 70 dollars, then find the greatest whole number of hours.
3 marks
$h \leq 7.5$ ; 8 whole hours would exceed the budget.
$25 + 6 h \leq 70$ ; 7 whole hours.
A recipe uses 3 units of flour and 2 units of sugar per batch A, or 1 unit of flour and 3 units of sugar per batch B. Exact totals of 17 flour units and 16 sugar units are used. Find the batch counts.
4 marks
$3 a + b = 17$ , $2 a + 3 b = 16$ ; substitution gives $7 a = 35$ .
5 batches of A and 2 batches of B.
A model gives 7.5 buses for transporting a group. Explain why this is not a directly usable bus count and what extra information is needed.
3 marks
Rounding down may leave passengers behind; do not round without checking the capacity constraint.
Whole buses are required. The capacity per bus and whether all buses must have the same capacity are needed to choose a feasible whole-number plan.
A council must buy at least 12 benches and tables in total. Benches cost 80 dollars and tables cost 120 dollars; the budget is 1200 dollars. Define variables, write constraints and test 9 benches with 3 tables.
4 marks
It provides 12 items.
Its cost is $720 + 360 = 1080$ dollars.
$b + t \geq 12$ , $80 b + 120 t \leq 1200$ , nonnegative integer $b, t$ ; the plan is feasible.

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian Mathematics Curriculum

Year 10 simultaneous equations and inequalities with graphical, algebraic and practical interpretations.

Official curriculum source ↗

In this lesson

Understand the idea.

Build a model from the information

Interpret, not just calculate

See the method in action.

Ticket prices and attendance

Define the counts and form both totals.

Substitute the count relation into the cost equation.

Solve for the adult count.

Find the child count.

Check both totals and the count domain.

Check a proposed plan

Write all constraints.

Check the labour requirement for the plan.

Check the total output.

Check the count domain and conclude.

Try it yourself.

Check your understanding

Keep practising