Simultaneous linear and non-linear equations
Use a line to substitute into a curve and find every allowed intersection.
Extended only. 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 quadratic equation.
- Substitute a complete expression into a square.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
After substituting y = x + 2, what is y²?
What you will learn
- Substitute a linear relation into a quadratic relation.
- Find up to two intersection pairs.
- Check pairs and interpret contextual restrictions.
One substitution reduces two unknowns to one
For y = x + 1 and x² + y² = 13, replace y by the entire bracket (x + 1). The equation becomes x² + (x + 1)² = 13, then 2x² + 2x − 12 = 0. Dividing by two gives x² + x − 6 = 0, so x = 2 or −3.
Recover y separately for each x: the pairs are (2, 3) and (−3, −2). Keeping the two pairs organised matters; mixing an x from one root with the y from the other gives a pair that does not lie on the line. Check both equations for each complete pair.
Intersections can be absent, repeated or distinct
A line may cross a quadratic curve twice, touch it once or miss it. After substitution, the resulting quadratic's discriminant tells which real case occurs. A repeated root corresponds to one contact point; it should not be listed twice as two different intersections.
For y = x + 2 and y = x², equate the two expressions for y: x² = x + 2. The factors (x − 2)(x + 1) give x = 2 or −1, and the pairs are (2, 4) and (−1, 1). Equating works because both expressions must describe the same y at an intersection.
Context selects among valid mathematical pairs
If x and y are lengths with y = x + 1 and xy = 12, substitution gives x(x + 1) = 12. The roots are 3 and −4. The algebraic pairs (3, 4) and (−4, −3) both satisfy the equations, but only (3, 4) fits positive lengths.
Graph readings are approximate unless coordinates are known exactly. An algebraic solution can check the graph and justify exact coordinates. When a question asks for a graphical method, show the curves and read the intersections instead of presenting only algebra. Always give the pair and its units if the problem represents measured quantities.
Pause and explain
If x = 2 or −1 and y = x + 2, which pairs follow?
Worked example
Solve y = x + 1 and x² + y² = 13.
Show the worked solution
- Substitute: x² + (x + 1)² = 13, giving x² + x − 6 = 0.
- Factorise (x + 3)(x − 2) = 0, so x = −3 or 2.
- Use y = x + 1 to get (−3, −2) and (2, 3). Each squared sum is 13.
Answer (x, y) = (−3, −2) or (2, 3)
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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 y = x + 2 and y = x².
Give me a hint
Equate the two expressions for the same y.
Compare my reasoning
- x² = x + 2 gives x² − x − 2 = 0.
- Factorise (x − 2)(x + 1) = 0, so x = 2 or −1.
- The matching y values are 4 and 1; the pairs satisfy both equations.
(2, 4) and (−1, 1)
Look for these in your work
- Reduced to a one-variable quadratic.
- Recovered the matching y for each root.
Solve y = 2x and x² + y² = 20.
Give me a hint
Square the whole substituted expression 2x.
Compare my reasoning
- x² + (2x)² = 20 gives 5x² = 20.
- x² = 4, so x = ±2.
- y = 2x gives (2, 4) and (−2, −4); both squared sums are 20.
(2, 4) and (−2, −4)
Look for these in your work
- Squared the coefficient as well as the variable.
- Included both real square roots.
Two positive numbers differ by one and their product is 12. Find both numbers using simultaneous equations.
Give me a hint
Let the larger number be y = x + 1.
Compare my reasoning
- Write y = x + 1 and xy = 12.
- Substitution gives x² + x − 12 = (x + 4)(x − 3) = 0.
- Positivity selects x = 3 and y = 4; the pair −4, −3 is excluded by the wording.
3 and 4
Look for these in your work
- Constructed both relationships.
- Applied the positive-number restriction after solving.
Common mistakes
- Squaring only part of a substituted expression.
- Finding x but not the corresponding y.
- Mixing coordinates from different roots.
What can you explain now?
Find the intersections of y = x and y = x² − 2.
Compare with the explanation
(−1, −1) and (2, 2)
Equate x = x² − 2, so (x − 2)(x + 1) = 0; use y = x for each pair.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Solve a line-and-curve pair, sketch their intersections and test every coordinate pair in both original equations.
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.