Greek Gymnasium Grade 3 · 60 minutes · 40 marks

Mathematics · Greek Gymnasium Grade 3: Expressions and systems

Algebraic expressions · Systems of linear equations

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.

Before you start

A complete original revision assessment of the two listed units only — not a full Panhellenic mock paper or whole-year assessment.
Timing is indicative. Show method, calculation and checking. In physics, include units and direction where required.
The indicative rubric accepts equivalent correct methods. Do not penalise the same arithmetic error again when subsequent working is consistent. This is not a diagnostic instrument or official grade.

Materials: Pencil, paper and ruler. Leave π exact where needed. · The numerical data do not require a calculator. This is not an official examination rule.

Algebraic expressions Systems of linear equations

Go to questions

Questions and marks

Work on paper first. Solutions stay closed until you choose to reveal them. Marks are for self-assessment, not automated or official grading.

1. Question 1

5 marks

Expand and simplify (7x+2)(x-6).

Worked solution and marking guide

$Multiply every term in the first bracket by every term in the second.$
$7x²-42x+2x-12$
$At x=0 both forms give -12.$

7x²-40x-12

Mark allocation

Multiply every term in the first bracket by every term in the second. — 1 marks
7x²−42x+2x−12 — 2 marks
At x=0 both forms give −12. — 2 marks

2. Question 2

5 marks

Factor x²-49.

Worked solution and marking guide

$Recognise a difference of two squares.$
$x²-7²=(x-7)(x+7)$
$Expanding makes the cross terms cancel.$

(x-7)(x+7)

Mark allocation

Recognise a difference of two squares. — 1 marks
x²−7²=(x−7)(x+7) — 2 marks
Expanding makes the cross terms cancel. — 2 marks

3. Question 3

5 marks

Simplify (x²-36)/(x-6) and state the restriction.

Worked solution and marking guide

$The original denominator requires x\neq6.$
$x²-36=(x-6)(x+6)$
$Cancel the common factor only for x\neq6.$

x+6, x\neq6

Mark allocation

The original denominator requires x≠6. — 1 marks
x²−36=(x−6)(x+6) — 2 marks
Cancel the common factor only for x≠6. — 2 marks

4. Question 4

5 marks

Simplify 7(x+2)-6(x-1).

Worked solution and marking guide

$Watch the negative sign before the second bracket.$
$7x+14-6x+6$
$The coefficient of x is 7-6=1.$

x+20

Mark allocation

Watch the negative sign before the second bracket. — 1 marks
7x+14−6x+6 — 2 marks
The coefficient of x is 7−6=1. — 2 marks

5. Question 5

5 marks

Solve x+y=13, x-y=1.

Worked solution and marking guide

$Add the equations to eliminate y.$
$2x=14 \Rightarrow x=7; y=13-7=6$
$Check the pair in both equations.$

x=7, y=6

Mark allocation

Add the equations to eliminate y. — 1 marks
2x=14 ⇒ x=7; y=13−7=6 — 2 marks
Check the pair in both equations. — 2 marks

6. Question 6

5 marks

Two ticket types cost €2 and €3. 13 tickets produced €32. Find how many of each type were sold.

Worked solution and marking guide

$Let x count €2 tickets and y count €3 tickets.$
$x+y=13; 2x+3y=32; 2x+2y=26; y=32-26=6; x=13-6=7$
$We subtracted twice the first equation from the second. The counts are non-negative integers and reproduce the total revenue.$

7 \times 2€, 6 \times 3€

Mark allocation

Let x count €2 tickets and y count €3 tickets. — 1 marks
x+y=13; 2x+3y=32; 2x+2y=26; y=32−26=6; x=13−6=7 — 2 marks
We subtracted twice the first equation from the second. The counts are non-negative integers and reproduce the total revenue. — 2 marks

7. Question 7

5 marks

How many solutions does x+y=7, 2x+2y=14 have? Give a general form.

Worked solution and marking guide

$The second equation is twice the first.$
$It adds no independent constraint.$
$y=7-x$

Infinitely many: (x, 7-x), x\inℝ.

Mark allocation

The second equation is twice the first. — 1 marks
It adds no independent constraint. — 2 marks
y=7−x — 2 marks

8. Question 8

5 marks

The pair (7,6) satisfies x+y=13. Is that enough to solve x-y=0 as well?

Worked solution and marking guide

$A system solution must satisfy every equation simultaneously.$
$7-6=1\neq0$
$Reject the pair even though it satisfies the first equation.$

No: it does not satisfy the second equation.

Mark allocation

A system solution must satisfy every equation simultaneously. — 1 marks
7−6=1≠0 — 2 marks
Reject the pair even though it satisfies the first equation. — 2 marks

Provenance and scope

Original authorship for YourFavTeacher with AI assistance.

Official sources were checked only for topic and level reference. No specific past-paper prompts, diagrams or solutions were copied or adapted. No external endorsement or licence to republish third-party papers is claimed.

Edition: 2026-09-06

Digital School — Mathematics, Gymnasium Grade 3 (new tab)

Record mistakes, revisit the relevant chapter and ask a tutor for an explanation if you need support.

Find a tutor in this subject All assessments

Greek Gymnasium Grade 3 · 60 minutes · 40 marks

Mathematics · Greek Gymnasium Grade 3: Expressions and systems

Algebraic expressions · Systems of linear equations

Before you start

A complete original revision assessment of the two listed units only — not a full Panhellenic mock paper or whole-year assessment.
Timing is indicative. Show method, calculation and checking. In physics, include units and direction where required.
The indicative rubric accepts equivalent correct methods. Do not penalise the same arithmetic error again when subsequent working is consistent. This is not a diagnostic instrument or official grade.

Materials: Pencil, paper and ruler. Leave π exact where needed. · The numerical data do not require a calculator. This is not an official examination rule.

Algebraic expressions Systems of linear equations

Go to questions

Questions and marks

Work on paper first. Solutions stay closed until you choose to reveal them. Marks are for self-assessment, not automated or official grading.

1. Question 1

5 marks

Expand and simplify (7x+2)(x-6).

Worked solution and marking guide

$Multiply every term in the first bracket by every term in the second.$
$7x²-42x+2x-12$
$At x=0 both forms give -12.$

7x²-40x-12

Mark allocation

Multiply every term in the first bracket by every term in the second. — 1 marks
7x²−42x+2x−12 — 2 marks
At x=0 both forms give −12. — 2 marks

2. Question 2

5 marks

Factor x²-49.

Worked solution and marking guide

$Recognise a difference of two squares.$
$x²-7²=(x-7)(x+7)$
$Expanding makes the cross terms cancel.$

(x-7)(x+7)

Mark allocation

Recognise a difference of two squares. — 1 marks
x²−7²=(x−7)(x+7) — 2 marks
Expanding makes the cross terms cancel. — 2 marks

3. Question 3

5 marks

Simplify (x²-36)/(x-6) and state the restriction.

Worked solution and marking guide

$The original denominator requires x\neq6.$
$x²-36=(x-6)(x+6)$
$Cancel the common factor only for x\neq6.$

x+6, x\neq6

Mark allocation

The original denominator requires x≠6. — 1 marks
x²−36=(x−6)(x+6) — 2 marks
Cancel the common factor only for x≠6. — 2 marks

4. Question 4

5 marks

Simplify 7(x+2)-6(x-1).

Worked solution and marking guide

$Watch the negative sign before the second bracket.$
$7x+14-6x+6$
$The coefficient of x is 7-6=1.$

x+20

Mark allocation

Watch the negative sign before the second bracket. — 1 marks
7x+14−6x+6 — 2 marks
The coefficient of x is 7−6=1. — 2 marks

5. Question 5

5 marks

Solve x+y=13, x-y=1.

Worked solution and marking guide

$Add the equations to eliminate y.$
$2x=14 \Rightarrow x=7; y=13-7=6$
$Check the pair in both equations.$

x=7, y=6

Mark allocation

Add the equations to eliminate y. — 1 marks
2x=14 ⇒ x=7; y=13−7=6 — 2 marks
Check the pair in both equations. — 2 marks

6. Question 6

5 marks

Two ticket types cost €2 and €3. 13 tickets produced €32. Find how many of each type were sold.

Worked solution and marking guide

$Let x count €2 tickets and y count €3 tickets.$
$x+y=13; 2x+3y=32; 2x+2y=26; y=32-26=6; x=13-6=7$
$We subtracted twice the first equation from the second. The counts are non-negative integers and reproduce the total revenue.$

7 \times 2€, 6 \times 3€

Mark allocation

Let x count €2 tickets and y count €3 tickets. — 1 marks
x+y=13; 2x+3y=32; 2x+2y=26; y=32−26=6; x=13−6=7 — 2 marks
We subtracted twice the first equation from the second. The counts are non-negative integers and reproduce the total revenue. — 2 marks

7. Question 7

5 marks

How many solutions does x+y=7, 2x+2y=14 have? Give a general form.

Worked solution and marking guide

$The second equation is twice the first.$
$It adds no independent constraint.$
$y=7-x$

Infinitely many: (x, 7-x), x\inℝ.

Mark allocation

The second equation is twice the first. — 1 marks
It adds no independent constraint. — 2 marks
y=7−x — 2 marks

8. Question 8

5 marks

The pair (7,6) satisfies x+y=13. Is that enough to solve x-y=0 as well?

Worked solution and marking guide

$A system solution must satisfy every equation simultaneously.$
$7-6=1\neq0$
$Reject the pair even though it satisfies the first equation.$

No: it does not satisfy the second equation.

Mark allocation

A system solution must satisfy every equation simultaneously. — 1 marks
7−6=1≠0 — 2 marks
Reject the pair even though it satisfies the first equation. — 2 marks

Provenance and scope

Original authorship for YourFavTeacher with AI assistance.

Edition: 2026-09-06

Digital School — Mathematics, Gymnasium Grade 3 (new tab)

Record mistakes, revisit the relevant chapter and ask a tutor for an explanation if you need support.

Find a tutor in this subject All assessments