Solve simultaneous linear and quadratic equations
Substitute or eliminate to find all intersection points of two equations and check each ordered pair.
Section 5 · Simultaneous equations
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which method is efficient when one equation already gives y = ...?
What you will learn
- Solve simultaneous equations in two unknowns by substitution or elimination
- Find all intersections of a line and a curve
- Check ordered pairs in both original equations
Substitution turns the system into one equation
When one equation is linear, rearrange it for x or y and substitute into the other. A line and a quadratic may have zero, one or two intersection points.
After solving, substitute each candidate back into the linear relation to recover the other coordinate. Report ordered pairs, not two unpaired lists.
Check every intersection in both equations
For x + y = 5 and xy = 6, use y = 5 − x. Then x(5 − x) = 6, so x² − 5x + 6 = 0 and x = 2 or 3.
The paired solutions are (2, 3) and (3, 2). Each pair satisfies both equations; keeping the pairs prevents mismatching coordinates.
Pause and explain
Why must each x-root be substituted into y = x + 2?
Worked example
Solve y = x + 2 and x² + y² = 10.
Show the worked solution
- Substitute y = x + 2 into the second equation.
- x² + (x + 2)² = 10 gives x² + 2x − 3 = 0.
- Factor to x = 1 or x = −3; use y = x + 2 to find each y-value.
Answer (x, y) = (1, 3) or (−3, −1).
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 y = 2x − 1 and x + y = 8.
Give me a hint
Substitute the first expression for y into the second equation.
Compare my reasoning
- x + (2x − 1) = 8.
- 3x = 9, so x = 3.
- Then y = 2(3) − 1 = 5.
(x, y) = (3, 5).
Look for these in your work
- Substitution correct
- x solved
- Ordered pair checked
Solve x + y = 5 and xy = 6.
Give me a hint
Write one variable in terms of the other, then factor the quadratic.
Compare my reasoning
- Let y = 5 − x, so x(5 − x) = 6.
- Rearrange to x² − 5x + 6 = 0 = (x − 2)(x − 3).
- Pair x = 2 with y = 3 and x = 3 with y = 2.
(2, 3) and (3, 2).
Look for these in your work
- Quadratic formed correctly
- Both roots found
- Pairs satisfy the original system
A line y = 4 − x meets a circle x² + y² = 10. Find both points and verify whether the line is tangent or a secant.
Give me a hint
Substitute the line into the circle and inspect the number of distinct roots.
Compare my reasoning
- x² + (4 − x)² = 10 gives x² − 4x + 3 = 0.
- Factor: (x − 1)(x − 3) = 0.
- The points are (1, 3) and (3, 1), so there are two intersections.
Two points, (1, 3) and (3, 1); the line is a secant, not a tangent.
Look for these in your work
- Equation substituted
- Both pairs computed
- Intersection type inferred
Common mistakes
- Squaring or expanding without brackets
- Finding x-values but not their corresponding y-values
- Checking only one equation
What can you explain now?
If a system's substitution gives a quadratic with two roots, what do those roots represent?
Compare with the explanation
Possible coordinates at two intersections
Each root must be paired with its corresponding value in the other variable.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Describe substitution for a simultaneous linear–quadratic system and the check for each ordered pair.
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.