Simultaneous equations from word problems
Translate weight, ticket and money stories into a pair of linear equations, then solve.
Key formulas
Define clearly
Write “Let x be …, let y be …” with units. Example pattern: Nancy weighs x kg, Ivan weighs y kg; total 70 kg and 3x = 2y produce the system x + y = 70, 3x = 2y (numbers varied in practice).
Check independence
The two equations must not be the same information twice. Totals, differences, and “times as many” usually give independent lines.
Solve and interpret
After finding x and y, check both original sentences. Reject negative lengths or impossible counts unless the context allows them.
Exam habit
In MCQ, match the equation pair before solving — often the question only asks which system is correct.
Worked example
Stamps problem
Problem. Amy and Ben have 90 stamps in total. Amy has 12 more than Ben. How many does each have?
Solution. Let Ben have b, Amy b+12. Then 2b+12 = 90 ⇒ 2b = 78 ⇒ b = 39. Amy has 51.
Review quiz
1 / 4Sum 90, Amy 12 more than Ben. Ben has
Practice questions
- Amy and Ben have 90 stamps in total. Amy has 12 more than Ben. Find how many each has.
- 2 adult tickets and 3 child tickets cost $210. 1 adult and 2 child tickets cost $120. Find each ticket price.
- x + y = 50 and 2x − y = 10. Solve for x and y.
- The sum of two numbers is 48 and three times the smaller equals the larger. Find the numbers.
Study notes
Substitution or elimination?
Use substitution when a variable is already isolated; otherwise elimination is often cleaner.
What if I get 0 = 5?
No solution — check the modelling of the word problem.