# Simultaneous Equations

> CIE IGCSE Additional Mathematics · CIE IGCSE Add Maths 0606
> Source: https://www.owlsprep.com/study/cie-0606-u5-overview/
> Weight: 5-7% of total exam marks

This unit covers solving pairs of simultaneous equations in two unknowns where at least one equation is non-linear (and sometimes both), a core algebraic skill tested explicitly in CIE IGCSE Additional Mathematics exams for both pure and applied problem types.

**Prerequisites:** Solving linear equations; Quadratic expansion, factorisation and solving

## Learning objectives

- Solve pairs of simultaneous equations in two unknowns where one or both equations are non-linear (quadratic, rational, circular, or containing an $xy$ term), using substitution or elimination
- Interpret solutions to simultaneous equations as intersection points of corresponding line and curve graphs
- Apply simultaneous equation solving to real-world and geometric context problems, rejecting any non-sensical solutions

## Unit at a glance

The unit follows a structured learning path focused on the specific simultaneous equation type required for the 0606 syllabus. You will first learn the substitution method to combine equations into a single solvable quadratic, then apply this skill to context-based and graphical interpretation problems.

This unit contains the following core sub-topic:
- [Simultaneous Equations (One or Both Non-Linear)](https://www.owlsprep.com/study/cie-0606-u5-simultaneous-equations/) — Master substitution and elimination to solve simultaneous equations where one or both equations are non-linear, including graphical interpretation and real-world context applications.

## Common pitfalls

- **Wrong:** Forgetting to substitute solved x/y values back into the original linear equation to find the corresponding second variable
  - Why it fails: This leaves solutions incomplete, leading to lost marks even if you correctly solved the intermediate quadratic equation
  - Correct: Always substitute each quadratic root back into the rearranged linear equation to get full (x,y) solution pairs, then verify pairs satisfy both original equations
- **Wrong:** Making rearrangement errors when isolating a variable from the linear equation before substitution
  - Why it fails: Small arithmetic errors during rearrangement propagate through the rest of your working, leading to incorrect quadratic roots and final solutions
  - Correct: Double-check your linear equation rearrangement before substituting it into the non-linear equation to catch avoidable calculation errors early

## Cheatsheet

| Key Concept/Formula | Description |
| --- | --- |
| Substitution Step | Rearrange the linear equation to make $x$ or $y$ the subject, then substitute this expression into the non-linear equation |
| Quadratic Form | After substitution, you will get a quadratic of form $ax^2 + bx + c = 0$, solvable via factorisation, completing the square, or quadratic formula |
| Solution Pairs | Each root of the resulting quadratic corresponds to one (x,y) solution pair for the original simultaneous equations |
| Graphical Interpretation | Solutions correspond to the intersection points of the line (linear equation) and curve (non-linear equation) on a coordinate plane |
| Context Problem Rule | Reject any solution that is impossible in the problem context (e.g. negative length, negative time) even if it satisfies the equations |

## What's next

Begin your learning for this unit by accessing the core sub-topic below, which includes step-by-step annotated examples and exam-style practice problems for one linear/one non-linear simultaneous equations. Once you have mastered this unit, you will progress to the next algebra unit covering linear and quadratic inequalities.

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