Learning objectives
- I can explain and apply: Substitution and elimination.
- I can explain and apply: Graphical interpretation.
- I can explain and apply: Modelling and verification.
Greek Gymnasium Grade 3 / Mathematics / Two-unit revision pilot
Systems of linear equations
Systems of linear equations: structured theory, worked examples, answered practice, and a mastery checklist for Greek Gymnasium Grade 3.
CHAPTER PLAN
Learn, practise, and check your progress
Follow the steps in order or jump directly to the part you need.
—/5
completed
Estimated active study time
156 minutes
1. Understand
Objectives, key ideas, and structured theory.
Open →2. Follow the method
Worked examples that explain every step.
Open →3. Practise
Graded tasks with hints and answer guidance.
Open →4. Consolidate
Assignments from core work to challenge.
Open →5. Check
A timed, marked test with solution guidance.
Open →
Progress is stored only in this browser and does not require AI credits.
Key ideas and checks
- Verify in both original equations.
- 0=1 indicates an inconsistent system.
- 0=0 may leave a free variable.
Unit
Core theory
The essential chapter ideas in a clear sequence before practice.
Understanding the concept
A system's solution is an ordered pair satisfying both equations simultaneously. Each equation represents a line: one intersection gives one solution, distinct parallel lines give none, and coincident lines give infinitely many.
Method and conditions
For substitution, isolate one unknown and replace it with the complete equivalent expression in brackets. For elimination, scale entire equations so one unknown has opposite coefficients, then add.
Checking and coverage boundaries
In a word problem, define what each unknown counts and write the two constraints separately. After solving, check units, non-negativity and integer values when counting objects.
Original pilot material — human educator review pending
Written specifically for YourFavTeacher with AI assistance, without copying past-paper prompts. Technical and numerical checks do not replace review by a human educator. It is not an official paper or endorsed by an examination body.
Mathematics
Worked examples
Follow the method step by step and check why every step is valid.
Worked example 1
Solve x+y=11, x−y=1.
- 1Add the equations to eliminate y.
- 22x=12 ⇒ x=6; y=11−6=5
- 3Check the pair in both equations.
x=6, y=5
Worked example 2
How many solutions does x+y=6, 2x+2y=12 have? Give a general form.
- 1The second equation is twice the first.
- 2It adds no independent constraint.
- 3y=6−x
Infinitely many: (x, 6−x), x∈ℝ.
Worked example 3
The pair (6,5) satisfies x+y=11. Is that enough to solve x−y=0 as well?
- 1A system solution must satisfy every equation simultaneously.
- 26−5=1≠0
- 3Reject the pair even though it satisfies the first equation.
No: it does not satisfy the second equation.
Systems of linear equations
Practice with answers
Eight graded tasks from core fluency to exam-style application. Work independently before opening a hint or answer.
Core fluency6 minutes
Solve x+y=5, x−y=1.
Hint
Add the equations to eliminate y.
Answer guide
Add the equations to eliminate y. 2x=6 ⇒ x=3; y=5−3=2 Check the pair in both equations. x=3, y=2
Core fluency6 minutes
Solve y=2x−3, x+y=6.
Hint
Substitute the first expression for y into the second equation.
Answer guide
Substitute the first expression for y into the second equation. x+2x−3=6 ⇒ 3x=9 ⇒ x=3; y=3 Both sides of the second equation equal 6. x=3, y=3
Core fluency6 minutes
Two ticket types cost €2 and €3. 5 tickets produced €12. Find how many of each type were sold.
Hint
Let x count €2 tickets and y count €3 tickets.
Answer guide
Let x count €2 tickets and y count €3 tickets. x+y=5; 2x+3y=12; 2x+2y=10; y=12−10=2; x=5−2=3 We subtracted twice the first equation from the second. The counts are non-negative integers and reproduce the total revenue. 3 × 2€, 2 × 3€
Application6 minutes
Does x+y=3, 2x+2y=7 have a solution?
Hint
Double the first equation.
Answer guide
Double the first equation. 2x+2y=6 The second demands a different total, 7, for the same expression. Inconsistent system: no solution.
Application6 minutes
How many solutions does x+y=3, 2x+2y=6 have? Give a general form.
Hint
The second equation is twice the first.
Answer guide
The second equation is twice the first. It adds no independent constraint. y=3−x Infinitely many: (x, 3−x), x∈ℝ.
Application6 minutes
Solve 2x+y=8, x+2y=7.
Hint
Double the second equation and subtract the first.
Answer guide
Double the second equation and subtract the first. 3y=6 ⇒ y=2; x=3 Check: 2·3+2=8 and 3+2·2=7. x=3, y=2
Reasoning6 minutes
The lines y=x+2 and y=−x+8 intersect. Find the intersection.
Hint
At the intersection the two expressions for y are equal.
Answer guide
At the intersection the two expressions for y are equal. x+2=−x+8 ⇒ 2x=6 ⇒ x=3; y=5 The point must lie on both lines. (3, 5)
Reasoning6 minutes
The pair (3,2) satisfies x+y=5. Is that enough to solve x−y=0 as well?
Hint
A system solution must satisfy every equation simultaneously.
Answer guide
A system solution must satisfy every equation simultaneously. 3−2=1≠0 Reject the pair even though it satisfies the first equation. No: it does not satisfy the second equation.
Common mistakes
- Checking only one equation.
- Scaling only one term during elimination.
- Treating 0=0 as the unique solution (0,0).
Mastery check
- I can solve independently and check: Substitution and elimination.
- I can solve independently and check: Graphical interpretation.
- I can solve independently and check: Modelling and verification.
Systems of linear equations
Chapter homework
Six distinct assignments from core fluency to challenge, each with an estimated time, hint, and answer guide.
Core8 minutes
Solve x+y=7, x−y=1.
Hint
Add the equations to eliminate y.
Answer guide
Add the equations to eliminate y. 2x=8 ⇒ x=4; y=7−4=3 Check the pair in both equations. x=4, y=3
Core8 minutes
Solve y=2x−4, x+y=8.
Hint
Substitute the first expression for y into the second equation.
Answer guide
Substitute the first expression for y into the second equation. x+2x−4=8 ⇒ 3x=12 ⇒ x=4; y=4 Both sides of the second equation equal 8. x=4, y=4
Stretch8 minutes
Two ticket types cost €2 and €3. 7 tickets produced €17. Find how many of each type were sold.
Hint
Let x count €2 tickets and y count €3 tickets.
Answer guide
Let x count €2 tickets and y count €3 tickets. x+y=7; 2x+3y=17; 2x+2y=14; y=17−14=3; x=7−3=4 We subtracted twice the first equation from the second. The counts are non-negative integers and reproduce the total revenue. 4 × 2€, 3 × 3€
Stretch8 minutes
Does x+y=4, 2x+2y=9 have a solution?
Hint
Double the first equation.
Answer guide
Double the first equation. 2x+2y=8 The second demands a different total, 9, for the same expression. Inconsistent system: no solution.
Challenge8 minutes
How many solutions does x+y=4, 2x+2y=8 have? Give a general form.
Hint
The second equation is twice the first.
Answer guide
The second equation is twice the first. It adds no independent constraint. y=4−x Infinitely many: (x, 4−x), x∈ℝ.
Challenge8 minutes
Solve 2x+y=11, x+2y=10.
Hint
Double the second equation and subtract the first.
Answer guide
Double the second equation and subtract the first. 3y=9 ⇒ y=3; x=4 Check: 2·4+3=11 and 4+2·3=10. x=4, y=3
45 minutes / 40 marks
Full chapter test
A timed, full-mark self-assessment with model-answer guidance.
Show working, units and a final check. Equivalent correct methods are accepted. Timing is a revision guide, not official examination conditions.
1. Solve x+y=9, x−y=1.
5 marksAnswer guide
Add the equations to eliminate y. 2x=10 ⇒ x=5; y=9−5=4 Check the pair in both equations. x=5, y=4
2. Solve y=2x−5, x+y=10.
5 marksAnswer guide
Substitute the first expression for y into the second equation. x+2x−5=10 ⇒ 3x=15 ⇒ x=5; y=5 Both sides of the second equation equal 10. x=5, y=5
3. Two ticket types cost €2 and €3. 9 tickets produced €22. Find how many of each type were sold.
5 marksAnswer guide
Let x count €2 tickets and y count €3 tickets. x+y=9; 2x+3y=22; 2x+2y=18; y=22−18=4; x=9−4=5 We subtracted twice the first equation from the second. The counts are non-negative integers and reproduce the total revenue. 5 × 2€, 4 × 3€
4. Does x+y=5, 2x+2y=11 have a solution?
5 marksAnswer guide
Double the first equation. 2x+2y=10 The second demands a different total, 11, for the same expression. Inconsistent system: no solution.
5. How many solutions does x+y=5, 2x+2y=10 have? Give a general form.
5 marksAnswer guide
The second equation is twice the first. It adds no independent constraint. y=5−x Infinitely many: (x, 5−x), x∈ℝ.
6. Solve 2x+y=14, x+2y=13.
5 marksAnswer guide
Double the second equation and subtract the first. 3y=12 ⇒ y=4; x=5 Check: 2·5+4=14 and 5+2·4=13. x=5, y=4
7. The lines y=x+4 and y=−x+14 intersect. Find the intersection.
5 marksAnswer guide
At the intersection the two expressions for y are equal. x+4=−x+14 ⇒ 2x=10 ⇒ x=5; y=9 The point must lie on both lines. (5, 9)
8. The pair (5,4) satisfies x+y=9. Is that enough to solve x−y=0 as well?
5 marksAnswer guide
A system solution must satisfy every equation simultaneously. 5−4=1≠0 Reject the pair even though it satisfies the first equation. No: it does not satisfy the second equation.
Unit
Official sources and verification
Curriculum reference sources. Always confirm the teaching sequence with the school and tutor.
Mathematics
A bridge year into Lyceum: students need confidence in algebraic manipulation and basic proof thinking.
What this chapter covers
The structure follows the official textbook layout and is used to organise study.
Substitution and elimination
Graphical interpretation
Modelling and verification
Where to focus
The areas that usually create mistakes or need extra revision.
I can solve independently and check: Substitution and elimination.
I can solve independently and check: Graphical interpretation.
I can solve independently and check: Modelling and verification.
Sources, daily material, and resources
Where to start: textbook, daily material, PDFs, videos, and worked examples.
Start from the official textbook or specification referenced on the subject page.
Use the notes and examples as support, not as a replacement for the official syllabus.
Check the current syllabus version before exam preparation.
Practice by subtopic
Targeted practice before full tests so coverage is clear.
Solve x+y=5, x−y=1.
Solve y=2x−3, x+y=6.
Two ticket types cost €2 and €3. 5 tickets produced €12. Find how many of each type were sold.
Does x+y=3, 2x+2y=7 have a solution?
How many solutions does x+y=3, 2x+2y=6 have? Give a general form.
Solve 2x+y=8, x+2y=7.
The lines y=x+2 and y=−x+8 intersect. Find the intersection.
The pair (3,2) satisfies x+y=5. Is that enough to solve x−y=0 as well?
Mocks and progress checks
How to measure progress in this chapter and when it enters a cumulative mock.
Start with untimed practice by subtopic.
Move to a short timed checkpoint only after completing the mastery checklist.
Record each error with the correct method and revisit it after 48 hours.
Next step
What to do after finishing the chapter and how it connects to the next unit.
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
Note: for the official examinable syllabus of each school year, always confirm with the school, tutor, and current Ministry/IEP announcements.