Practical simultaneous/inequality models and checks | Year 10

01 / LEARN

The idea, step by step

Build a model from the information

Define each variable and its units before writing equations. A count total and a cost total often provide two different equations. Exact requirements use equality; a maximum or minimum uses an inequality. Do not confuse a price per item with the number of items.

Interpret, not just calculate

After solving, substitute into the original totals. Reject negative counts or fractional items when whole objects are required. A mathematical model may omit fees, waste or time limits; state any assumption that could affect a real decision. A feasible pair need not be the only feasible pair.

A common trap

Treating an algebraic solution with a negative number of tickets as a valid sale.

Instead: The original situation imposes a domain. Counts must be nonnegative whole numbers, so an invalid result may reveal inconsistent data or an unsuitable model.

02 / WORKED EXAMPLES

See the thinking

Example 1

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?

Let $a, c$ be the ticket counts. Then $a + c = 13$ and $10 a + 6 c = 98$ .
Use $c = 13 - a$ : $10 a + 6 (13 - a) = 98$ .
$4 a = 20$ , so $a = 5, c = 8$ .
Check the count and the cost: 5+8=13 and 50+48=98.

5 adult tickets and 8 child tickets.

Example 2

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?

The constraints are $3 a + 2 b \leq 30$ , $a + b \geq 8$ , with nonnegative integer counts.
The plan uses 18+8=26 hours and produces 10 items.
Both requirements are satisfied.

Yes: it uses 26 hours and produces 10 items.

Check your understanding

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

Show answer

Notebook: 4 dollars; pen: 3 dollars.

Subtract the second total from the first to find the notebook price, then substitute.

03 / WORKSHEET

Your turn

6 questions · 20 marks · Work at your own pace.

Worksheet PDF 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.

Checking saved progress…

Ready for the topic test?

Review the chapter, then choose a practice test.

Topic review

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 ↗

Back to Linear equations and inequalities