Grade 11 Maths · Algebra · Paper 1

Simultaneous Equations Grade 11

Real Simultaneous Equations past paper questions and answers from 2024 to 2025, with each memo split into the ticks examiners give. Try one, mark yourself, and see exactly which skill cost you marks.

past paper questions: 23 past paper questions
skills: 3 skills
videos: 3 videos

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Question 1.3 · March 2025 · Provincial · 6 marks

1.3 Solve simultaneously for

x

and

y

:\n

2 x - y = 3

and

x^{2} - 15 = - 5 x y - y^{2}

.(6)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

Simultaneous Equations skills

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More Simultaneous Equations Grade 11 questions and answers

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4. Question 1.2 · November 2025 · Paper 1 · Provincial · 6 marks
1.2 Solve simultaneously for $x$ and $y$ :
$x - 3 y = 1$
$x^{2} - 2 x y + 9 y^{2} = 17$ (6)
Show the memo (6 ticks)Memo
1. ✓
  Make $x$ the subject $x - 3 y = 1 (1)$
  $x^{2} - 2 x y + 9 y^{2} = 17 (2)$
  From (1): $x = 3 y + 1 (3)$
  1 mark · Change the subject of a formula
2. ✓
  Substitution(3) into (2):
  $(3 y + 1)^{2} - 2 y (3 y + 1) + 9 y^{2} = 17$
  1 mark · Solve linear and non-linear simultaneous equations
3. ✓
  Standard form $9 y^{2} + 6 y + 1 - 6 y^{2} - 2 y + 9 y^{2} - 17 = 0$
  $12 y^{2} + 4 y - 16 = 0$
  $3 y^{2} + y - 4 = 0$
  1 mark · Write a quadratic in standard form
4. ✓
  Factors $(y - 1) (3 y + 4) = 0$
  1 mark · Factorise trinomials with a leading coefficient
5. ✓
  $y$ -values $y = 1$ or $y = - \frac{4}{3}$
  1 mark · Solve a quadratic by factorising
6. ✓
  $x$ -values $x = 4$ or $x = - 3$
  1 mark · Solve linear and non-linear simultaneous equations
5. Question 1.4 · June 2024 · Paper 1 · Provincial · 6 marks
1.4 Solve simultaneously for $x$ and $y$ : $2 x = y + 2$ and $2 x^{2} = 2 - y^{2}$ (6)
Show the memo (6 ticks)Memo
1. ✓
  Make variable subject of formula $y = 2 x - 2$
  1 mark · Change the subject of a formula
2. ✓
  Substitute into quadratic equation $2 x^{2} = 2 - (2 x - 2)^{2}$
  $2 x^{2} = 2 - (4 x^{2} - 8 x + 4)$
  1 mark · Solve linear and non-linear simultaneous equations
3. ✓
  Standard form $6 x^{2} - 8 x + 2 = 0$
  $3 x^{2} - 4 x + 1 = 0$
  1 mark · Write a quadratic in standard form
4. ✓
  Factorise $(3 x - 1) (x - 1) = 0$
  1 mark · Factorise trinomials with a leading coefficient
5. ✓
  Solve for x $x = \frac{1}{3}$ or $x = 1$
  1 mark · Solve a quadratic by factorising
6. ✓
  Solve for yIf $x = \frac{1}{3}$ , $y = 2 (\frac{1}{3}) - 2 = - \frac{4}{3}$
  If $x = 1$ , $y = 2 (1) - 2 = 0$
  1 mark · Solve linear and non-linear simultaneous equations
6. Question 1.3 · March 2024 · Paper 1 · Provincial · 6 marks
1.3 Solve simultaneously for $x$ and $y$ :
$4 x + y = 7$ and $3 x^{2} + 2 x y = y^{2}$ (6)
Show the memo (5 ticks)Memo
1. ✓
  Make y the subject of the linear equation $y = 7 - 4 x$
  1 mark · Change the subject of a formula
2. ✓
  Substitute into quadratic equation $3 x^{2} + 2 x (7 - 4 x) = (7 - 4 x)^{2}$
  1 mark · Solve linear and non-linear simultaneous equations
3. ✓
  Simplify to standard form $3 x^{2} + 14 x - 8 x^{2} = 49 - 56 x + 16 x^{2}$
  $21 x^{2} - 70 x + 49 = 0$
  $3 x^{2} - 10 x + 7 = 0$
  1 mark · Write a quadratic in standard form
4. ✓
  Factorise and solve for x $(3 x - 7) (x - 1) = 0$
  $x = \frac{7}{3}$ or $x = 1$
  1 mark · Solve a quadratic by factorising
5. ✓
  Solve for yIf $x = \frac{7}{3}$ , $y = 7 - 4 (\frac{7}{3}) = - \frac{7}{3}$
  If $x = 1$ , $y = 7 - 4 (1) = 3$
  2 marks · Solve linear and non-linear simultaneous equations
7. Question 1.3 · November 2024 · Paper 1 · Provincial · 6 marks
1.3 Solve for $x$ and $y$ simultaneously:

$- 3 y = 1 - x$ and $x^{2} - 2 x y - y^{2} - 7 = 0$ (6)
Show the memo (6 ticks)Memo
1. ✓
  $x = 1 + 3 y$ $- 3 y = 1 - x$ ...(1)
  $x^{2} - 2 x y - y^{2} - 7 = 0$ ...(2)
  From (1): $x = 1 + 3 y$ ...(3)
  1 mark · Change the subject of a formula
2. ✓
  Substitution(3) into (2):
  $(1 + 3 y)^{2} - 2 y (1 + 3 y) - y^{2} - 7 = 0$
  1 mark · Solve linear and non-linear simultaneous equations
3. ✓
  Standard form $1 + 6 y + 9 y^{2} - 2 y - 6 y^{2} - y^{2} - 7 = 0$
  $2 y^{2} - 4 y - 6 = 0$
  1 mark · Write a quadratic in standard form
4. ✓
  Factors $y^{2} - 2 y - 3 = 0$
  $(y - 3) (y + 1) = 0$
  1 mark · Factorise a trinomial
5. ✓
  $y$ -values $y = 3$ or $y = - 1$
  1 mark · Solve a quadratic by factorising
6. ✓
  $x$ -values $x = 10$ or $x = - 2$
  1 mark · Solve linear and non-linear simultaneous equations
8. Question 2.3 · November 2024 · Paper 1 · Provincial · 5 marks
2.3 Calculate a possible value for $a^{2} + b^{2}$ , if it is given that $a$ and $b$ are rational numbers such that $a + b = 9 + 56$ .(5)
Show the memo (5 ticks)Memo
1. ✓
  Squaring both sides $(a + b)^{2} = (9 + 56)^{2}$
  $a + b + 2 ab = 9 + 2 \times 14$
  1 mark · Square a binomial
2. ✓
  $56 = 214$ $a + b + 2 ab = 9 + 214$
  but $56 = 214$
  1 mark · Simplify and combine surds
3. ✓
  $a + b = 9$ and $2 ab = 214$ $∴ a + b = 9$ and $2 ab = 214 \Rightarrow ab = 14$
  1 mark · Simplify and combine surds
4. ✓
  $a = 2$ or $b = 7$ (by inspection)by inspection:
  $a = 2$ or $b = 7$
  1 mark · Solve linear and non-linear simultaneous equations
5. ✓
  $a^{2} + b^{2} = 53$ $∴ a^{2} + b^{2} = (2)^{2} + (7)^{2} = 53$
  1 mark · Substitute into a formula
9. Question 4.5 · November 2024 · Paper 2 · Provincial · 5 marks
In the diagram below, P(-1 ; 3), R(3 ; -1) and S( $a$ ; $b$ ) are the vertices of $Δ PRS$ .
Point T lies on the perpendicular bisector of PR. $Diagram for this question$
4.5 If the area of $Δ PRS = 12$ square units and $a, b < 0$ , determine the coordinates of S.(5)
Show the memo (5 ticks)Memo
1. ✓
  Substitution in area formulaArea $Δ PRS = \frac{1}{2} PR . TS$
  $12 = \frac{1}{2} \times 42 \times (a - 1)^{2} + (b - 1)^{2}$
  1 mark · Calculate area or perimeter
2. ✓
  Square both sides $32 = (a - 1)^{2} + (b - 1)^{2}$
  $18 = (a - 1)^{2} + (b - 1)^{2}$
  1 mark · Solve a surd equation
3. ✓
  Substitute $a = b$ but $a = b$
  $∴ 18 = (a - 1)^{2} + (a - 1)^{2}$
  1 mark · Solve simultaneous equations by substitution
4. ✓
  Values of $a$ $18 = 2 (a - 1)^{2}$
  $9 = (a - 1)^{2}$
  $a - 1 = \pm 9$
  $a - 1 = - 3$ or $a - 1 = 3$
  or: $0 = a^{2} - 2 a - 8$ , $0 = (a - 4) (a + 2)$ , $a = 4$ / $a = - 2$
  1 mark · Solve a quadratic by factorising
5. ✓
  Coordinates of Sbut $a < 0$
  $∴ a - 1 = - 3$
  $∴ a = - 2$
  $∴ S (- 2; - 2)$
  1 mark · Reject solutions that do not fit
10. Question 10.4 · November 2024 · Paper 2 · Provincial · 6 marks
The circle is given below with TQA as a tangent. PS = SR and PQ = PR. $PQ ∥ SR$ .
SR is extended to A. $\hat{Q}_{1} = x$ and $\hat{Q}_{3} = y$ . $Diagram for this question$
10.4 Calculate the value of $x$ and $y$ .(6)
Show the memo (6 ticks)Memo
1. ✓
  S/R: angles on a straight line $\hat{Q}_{1} + \hat{Q}_{2} + \hat{Q}_{3} = 180°$ [ $∠$ s on a str line]
  1 mark · Use angles on a straight line
2. ✓
  $y = 180° - 2 x$ $x + x + y = 180°$
  $y = 180° - 2 x$ .....(1)
  1 mark · Use angles on a straight line
3. ✓
  S/R: angles in a triangle $\hat{P}_{1} + \hat{R}_{1} + \hat{S} = 180°$ [ $∠$ s in a $Δ$ ]
  1 mark · Use angle sum of a triangle
4. ✓
  $2 y = x$ $y + y + 180° - x = 180°$
  $2 y = x$ .... (2)
  1 mark · Use angle sum of a triangle
5. ✓
  $x$ -valueSubt (1) in (2)
  $2 (180° - 2 x) = x$
  $360° - 4 x = x$
  $360° = 5 x$
  $x = 72°$
  1 mark · Solve a linear equation
6. ✓
  $y$ -value $∴ y = 180° - 2 (72°)$
  $y = 36°$
  1 mark · Solve simultaneous equations by substitution

Simultaneous Equations Grade 11 videos

Short lessons from South African teachers first. Videos load only when you tap play, to save your data.

Literal Equations Grade 10 Maths Equations PART 1Miss Martins Maths and Science

Simultaneous Equations grade 10 | SubstitutionKevinmathscience

3) Simultaneous equations gr 11 | IntroKevinmathscience

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