Variables and equivalence
A variable represents a number or quantity. Every algebraic transformation must preserve the equivalence of the original expression or equation.
Greek Gymnasium Grade 3 / Mathematics / Μέρος Α: Άλγεβρα
Συστήματα γραμμικών εξισώσεων: structured theory, worked examples, answered practice, and a mastery checklist for Greek Gymnasium Grade 3.
Unit
The essential chapter ideas in a clear sequence before practice.
A variable represents a number or quantity. Every algebraic transformation must preserve the equivalence of the original expression or equation.
Write each transformation on a new line and apply the same valid operation to both sides. This keeps the solution easy to audit.
A final value is not a complete solution until it is substituted into the original relation. Verification catches sign and expansion errors.
Mathematics
Follow the method step by step and check why every step is valid.
Solve 3(x - 2) + 6 = 2x + 12.
Mathematics
A bridge year into Lyceum: students need confidence in algebraic manipulation and basic proof thinking.
The structure follows the official textbook layout and is used to organise study.
The areas that usually create mistakes or need extra revision.
x = 12
Before calculating, explain the key idea from “Γραμμική εξίσωση με δύο αγνώστους” and which conditions must be checked.
The answer should show not only which rule is used for “Γραμμική εξίσωση με δύο αγνώστους”, but also why it is valid here.
Συστήματα γραμμικών εξισώσεων
Try independently, use the hint if needed, then open the answer guide.
1. Explain the idea and give one correct foundation example for “Γραμμική εξίσωση με δύο αγνώστους”.
Write the givens, a useful representation or rule, and only then calculate.
A complete answer defines “Γραμμική εξίσωση με δύο αγνώστους”, shows equivalent steps, and includes a final check.
2. Solve an application and show every intermediate step for “Γραφική λύση”.
Write the givens, a useful representation or rule, and only then calculate.
A complete answer defines “Γραφική λύση”, shows equivalent steps, and includes a final check.
3. Compare a correct and an incorrect approach and justify the difference for “Μέθοδος αντικατάστασης”.
Write the givens, a useful representation or rule, and only then calculate.
A complete answer defines “Μέθοδος αντικατάστασης”, shows equivalent steps, and includes a final check.
4. Create a short exam-style question and check your answer for “Μέθοδος αντίθετων συντελεστών”.
Write the givens, a useful representation or rule, and only then calculate.
A complete answer defines “Μέθοδος αντίθετων συντελεστών”, shows equivalent steps, and includes a final check.
Where to start: textbook, daily material, PDFs, videos, and worked examples.
Targeted practice before full tests so coverage is clear.
How to measure progress in this chapter and when it enters a cumulative mock.
What to do after finishing the chapter and how it connects to the next unit.
Note: for the official examinable syllabus of each school year, always confirm with the school, tutor, and current Ministry/IEP announcements.