Forming and solving systems from a context
Turning words into equations
- Read the problem and decide what the unknowns are — define each with a letter and its meaning (e.g. let = number of adult tickets).
- Each separate fact in the problem becomes one equation.
- To pin down three unknowns you need three independent facts — so look for three.
- Watch for the different kinds of fact:
- a total (number of items, total cost, total mass),
- a rate or price applied to each unknown,
- a relationship between unknowns ("twice as many", "$5 more than").
Setting the system up
- Keep units consistent — dollars with dollars, kilograms with kilograms.
- Write each equation in the same variable order, so elimination is tidy.
- A "twice as many as " statement becomes , which is often the easiest to substitute first.
Solving and interpreting
- Solve the system by elimination or substitution.
- Interpret the answer back in context — with the right units and a check that it is sensible (you cannot buy tickets or half a person).
- Answer the actual question asked, which may be a single value or a combination of the unknowns.
For a Great Walk a tramper buys muesli bars, bags of scroggin and dried-meal pouches. She buys 12 items in total. Muesli bars cost $2, scroggin $3 and meal pouches $8 each, for a total of $42. She buys twice as many muesli bars as meal pouches. How many of each does she buy?
Step 1 — Define variables
Let = muesli bars, = scroggin bags, = meal pouches.
Step 2 — Form three equations
Step 3 — Substitute (3) into (1) and (2)
Into (1): , so .
Into (2):
Step 4 — Back-substitute
and .
Step 5 — Interpret in context
Test yourself
Practice by grade
One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.
Three whole numbers add to 15. The second is twice the first, and the third is 3 more than the first. Set up a system and find the numbers.
A café sells 100 hot drinks one morning — flat whites, long blacks and hot chocolates. Flat whites are $5, long blacks $4 and hot chocolates $4.50, taking $460 in total. There are three times as many flat whites as hot chocolates. How many of each were sold?
A fruit grower blends apple, feijoa and kiwifruit juice to make 60 litres of a mix. The mix must contain twice as much apple as kiwifruit. Apple juice is $3/L, feijoa $6/L and kiwifruit $9/L, and the 60 L blend must cost exactly $300. Find how many litres of each the blend contains, and comment on whether the recipe is sensible.