Linear simultaneous equations and modelling
Find the pair that satisfies two relationships at the same time, using elimination or substitution.
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Solve a linear equation in one unknown.
- Multiply a whole equation and substitute an expression.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is twice the entire equation x + 2y = 7?
What you will learn
- Construct two equations in two unknowns.
- Use elimination and substitution.
- Interpret unique, absent and non-unique intersections.
One pair must satisfy both equations
A solution of x + y = 8 alone could be (1, 7), (2, 6) or many other pairs. Adding x − y = 2 restricts the pair. Addition eliminates y to give 2x = 10, so x = 5 and y = 3. Both original equations must hold, not just the one used at the last step.
Graphically, each linear equation is a straight line and a unique simultaneous solution is their intersection. Parallel distinct lines have no solution. Equivalent equations describe the same line and have infinitely many common solutions. Algebra can reveal these cases as a contradiction or an identity after elimination.
Make a matching coefficient for elimination
For 2x + 3y = 17 and 3x − 2y = 6, multiply the first equation by two and the second by three. They become 4x + 6y = 34 and 9x − 6y = 18. Adding gives 13x = 52, so x = 4. Substitute to obtain y = 3.
Multiply every term, including the constant, when scaling an equation. If matching terms have equal signs, subtract; if their signs are opposite, add. Keep the two scaled equations on separate lines so signs remain visible. There is no requirement to eliminate x rather than y: choose the smaller convenient multipliers.
Substitute a whole expression
If y = 2x + 1 and x + y = 10, substitute to obtain x + (2x + 1) = 10. Solving gives x = 3 and y = 7. Brackets protect the replacement when a coefficient or minus sign acts on it. Substitution is especially useful when one variable is already isolated.
To model purchases, define the two unit prices before writing totals. If two adult tickets and three child tickets cost $39, write 2a + 3c = 39. Another independent purchase gives a second equation. Include units in the answer and check both totals, since reversing the two prices can accidentally fit only one relationship.
Pause and explain
For 3x + 2y = 12 and 5x − 2y = 4, which operation eliminates y?
Worked example
Solve 2x + 3y = 17 and 3x − 2y = 6.
Show the worked solution
- Scale by two and three respectively: 4x + 6y = 34 and 9x − 6y = 18.
- Add to get 13x = 52, hence x = 4.
- Substitute: 8 + 3y = 17, so y = 3. The second equation gives 12 − 6 = 6.
Answer x = 4, y = 3
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Solve x + y = 9 and x − y = 3.
Give me a hint
The y coefficients are already opposites.
Compare my reasoning
- Add the equations to get 2x = 12.
- Then x = 6; substitute in x + y = 9 to get y = 3.
- Check 6 − 3 = 3 in the second equation.
x = 6, y = 3
Look for these in your work
- Eliminated one variable before solving.
- Checked both equations.
Solve y = 3x − 2 and 2x + y = 13.
Give me a hint
Replace y in the second equation by the full expression.
Compare my reasoning
- 2x + (3x − 2) = 13.
- 5x = 15, so x = 3.
- y = 3 × 3 − 2 = 7; 2 × 3 + 7 = 13 checks the pair.
x = 3, y = 7
Look for these in your work
- Used the complete substituted expression.
- Recovered the second variable.
Two adult and three child tickets cost $39; three adult and one child ticket cost $34. Find each price.
Give me a hint
Let a and c be prices, then eliminate one from the two totals.
Compare my reasoning
- Write 2a + 3c = 39 and 3a + c = 34.
- Triple the second equation and subtract the first: 7a = 63, so a = 9.
- c = 34 − 27 = 7; both purchases check.
Adult $9; child $7
Look for these in your work
- Constructed independent equations from the purchases.
- Checked both monetary totals.
Common mistakes
- Scaling only the unknown terms.
- Subtracting equations with opposite coefficients when addition is needed.
- Reporting a pair that satisfies only one equation.
What can you explain now?
Solve x + 2y = 11 and 2x − y = 7.
Compare with the explanation
x = 5, y = 3
The first equation plus twice the second gives 5x = 25, so x = 5. Substitution gives y = 3; both equations check.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Solve the same simultaneous pair once by elimination and once by substitution. Explain the intersection the two methods identify.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.