Many real problems have two unknown amounts and two facts that connect them. Each fact becomes an equation, and solving the system gives both amounts.
A method for any word problem
- Name the unknowns. Say exactly what each letter stands for, such as a = number of adult tickets.
- Write one equation for each fact in the question.
- Solve the system by substitution or elimination.
- Answer the question in words, with units.
- Check the answer against the original wording, not just your equations.
Worked examples
Example 1: tickets
A school sold 30 tickets for a play. Adult tickets cost £8 and child tickets cost £5. The total was £195. How many of each were sold?
Let a = adult tickets and c = child tickets.
Number of tickets: a + c = 30. Money: 8a + 5c = 195.
From the first equation, c = 30 − a. Substitute: 8a + 5(30 − a) = 195, so 3a + 150 = 195, 3a = 45, and a = 15. Then c = 15.
15 adult and 15 child tickets. Check: 15 × £8 + 15 × £5 = £120 + £75 = £195 ✓.
Example 2: comparing two phone plans
Plan A costs £20 a month plus £2 per GB of data. Plan B costs £8 a month plus £5 per GB. For what amount of data do they cost the same?
Let g = gigabytes used and C = monthly cost in pounds.
Plan A: C = 2g + 20. Plan B: C = 5g + 8.
Set them equal: 2g + 20 = 5g + 8, so 12 = 3g and g = 4. Then C = 2 × 4 + 20 = 28.
At 4 GB both plans cost £28. Below 4 GB Plan B is cheaper, and above 4 GB Plan A is cheaper.
Example 2 is a good case for a graph. Each plan is a straight line, the slope is the price per GB, and the y-intercept is the fixed monthly charge. The crossing point is where the plans cost the same.
Example 3: two numbers
Two numbers add up to 25 and their difference is 7. Find the numbers.
Let x be the larger number and y the smaller: x + y = 25 and x − y = 7.
Add the equations: 2x = 32, so x = 16. Then y = 25 − 16 = 9.
The numbers are 16 and 9. Check: 16 + 9 = 25 ✓ and 16 − 9 = 7 ✓.
See Example 3 in the Intersection Finder →
Spotting the two equations
- “Total”, “altogether”, “sum” usually means adding the unknowns.
- A price or rate per item gives the coefficients, such as 8a for £8 per adult ticket.
- “Costs the same”, “equal”, “break even” means setting two expressions equal.
- If you only have one equation, look again: the question almost always hides a second fact.