This module covers pairs of equations that must be true at the same time. You solve two linear equations by elimination or substitution, and read what the answer means in real life. You also meet a line and a curve, and spot pairs with no solution.
The same skills return in graph work, inequalities and modelling.
Check the current Cambridge IGCSE Mathematics 0580 syllabus for the exact wording in your exam year, especially for the linear and quadratic pair. The methods stay the same across versions. Our Mathematics learning guide shows where this module sits among the others.
What should you already know?
You should be comfortable solving a single linear equation, expanding brackets and handling negative numbers. The earlier module on equations and formulas covers these. For the last lesson on a line and a curve you also need to factorise a quadratic.
An orienting example
Solve 2x + y = 8 and x − y = 1.
Step 1, elimination: the y terms have opposite signs, so add: 3x = 9, so x = 3.
Step 2, find y: 3 − y = 1, so y = 2.
Step 3, check: 2(3) + 2 = 8, which matches.
A second route, substitution: from the second equation, y = x − 1. Put that into the first: 2x + (x − 1) = 8, so 3x = 9 and x = 3 again.
Picture: each equation is a straight line, and (3, 2) is the one point where they cross. That single picture connects every lesson in this module.
In which order should you study it?
- Solve two linear equations by elimination: the main method, and the one that works when no letter is alone.
- Choose substitution when one variable is isolated: the shorter route when a rule such as y = 2x − 1 is given.
- Interpret an intersection in context: turns the numbers into a sentence that earns the mark.
- Solve a linear and quadratic pair where applicable: uses substitution and factorising together, and produces two points.
- Detect inconsistent simultaneous conditions: what a line like 0 = 3 means, and how to recognise parallel lines.
Then work through the mixed practice set. A steady pace is one lesson an evening, then the practice set at the weekend.
Which traps catch most students here?
- Multiplying only the left-hand side when scaling an equation.
- Adding when you should subtract, because the signs of the terms being removed were not checked.
- Dropping brackets when substituting an expression.
- Stopping at x and forgetting to find y.
- Pairing answers wrongly in a line and curve problem.
- Treating 0 = 3 as an equation instead of a sign that there is no solution.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Try each question on paper and write the same working you would in an exam. Method lines often earn marks even when the final value is wrong. The non-calculator working trainer lets you check arithmetic steps without reaching for a calculator.
When you get something wrong, use the routing table at the end of the practice set to return to the right lesson. Record the error in the mistake log, and retry a fresh question a few days later. The quadratic structure explorer helps you picture a line and a curve together.
If simultaneous equations still end in sign slips or a missing y, a teacher in online one-to-one Mathematics tuition can watch you solve one live and correct the habit.