Grade 11 Maths · Algebra

Solve linear and non-linear simultaneous equations

Solve a straight-line equation and a curved one together by substituting the line into the other (for example y = x + 1 into y = x² − 1).

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3) Simultaneous equations gr 11 | IntroKevinmathscience

Simultaneous Equations Grade 11 Equations Using SubstitutionMiss Martins Maths and Science

3e) Simultaneous equations gr 11 | TryKevinmathscience

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Question 1.2 · November 2024 · Paper 1 · National (DBE) · 5 marks

1.2 Solve for

x

and

y

simultaneously:

2 x + y = 3

and

y^{2} + x y = 2

(5)

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

More past paper questions: solve linear and non-linear simultaneous equations

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3. Question 1.2 · May/June 2025 · Paper 1 · National (DBE) · 6 marks
1.2 A rectangle having sides of $(y - 3)$ metres and $(x + 2)$ metres has a perimeter of $24$ metres and an area of $32$ square metres. Calculate the values of $x$ and $y$ .(6)
Show the memo (4 ticks)Memo
1. ✓
  Set up equations for area and perimeterArea: $(y - 3) (x + 2) = 32$
  $y x + 2 y - 3 x - 6 = 32$ ....(1)
  
  Perimeter: $2 (y - 3) + 2 (x + 2) = 24$
  $y + x - 1 = 12$
  $y = 13 - x$ ....(2)
  2 marks · Set up an expression or equation
2. ✓
  Substitute equation 2 into equation 1 $(13 - x) x + 2 (13 - x) - 3 x - 6 = 32$
  $13 x - x^{2} + 26 - 2 x - 3 x - 6 - 32 = 0$
  1 mark · Solve linear and non-linear simultaneous equations
3. ✓
  Write in standard form and factorise $- x^{2} + 8 x - 12 = 0$
  $x^{2} - 8 x + 12 = 0$
  $(x - 6) (x - 2) = 0$
  1 mark · Write a quadratic in standard form
4. ✓
  Determine values of x and y $x = 6$ or $x = 2$
  
  If $x = 6$ :
  $y = 13 - 6 = 7$
  
  If $x = 2$ :
  $y = 13 - 2 = 11$
  2 marks · Solve a quadratic by factorising
4. Question 1.4 · March 2021 · Provincial · 5 marks
1.4 Atlehang generates a sequence, which he claims that the sequence can be arithmetic and also be geometric if and only if the first term in the sequence is 1. Is he correct? Motivate your answer by showing all your calculations.(5)
Show the memo (3 ticks)Memo
1. ✓
  Set up equations for arithmetic and geometric sequencesArithmetic: $1; 1 + d; 1 + 2 d; ...$
  Geometric: $1; r; r^{2}; ...$
  $r = 1 + d$ ... (1)
  $r^{2} = 1 + 2 d$ ... (2)
  1 mark · Set up an expression or equation
2. ✓
  Substitute and solveSubstitute (1) into (2):
  $(1 + d)^{2} = 1 + 2 d$
  $1 + 2 d + d^{2} = 1 + 2 d$
  $d^{2} = 0$
  $d = 0$
  2 marks · Solve linear and non-linear simultaneous equations
3. ✓
  Determine r and concludeIf $d = 0$ , then $r = 1$ .
  Arithmetic sequence: $1; 1; 1; ...$
  Geometric sequence: $1; 1; 1; ...$
  He is correct.
  2 marks · Solve linear and non-linear simultaneous equations
5. Question 6.6 · May/June 2024 · Paper 1 · National (DBE) · 5 marks
The graphs of $f (x) = x^{2} - 2 x - 3$ and $g (x) = m x + c$ are drawn below. D and E are the $x$ -intercepts and P is the $y$ -intercept of $f$ . The turning point of $f$ is T $(1; - 4)$ . The graphs of $f$ and $g$ intersect at P and E.
6.6 Given: $k (x) = g (x) - n$ . Determine $n$ if $k$ is a tangent to $f$ .(5)
Show the memo (5 ticks)Memo
1. ✓
  Equate derivative of f to gradient of g $2 x - 2 = 1$
  1 mark · Find equation of a tangent
2. ✓
  Find x-coordinate of tangency $x = 1, 5$
  1 mark · Find equation of a tangent
3. ✓
  Find y-coordinate on f $y = (1, 5)^{2} - 2 (1, 5) - 3 = - 3, 75$
  1 mark · Find equation of a tangent
4. ✓
  Substitute point into k(x) equation $- 3, 75 = 1, 5 - 3 - n$
  1 mark · Find equation of a tangent
5. ✓
  Solve for n $n = 2, 25$
  1 mark · Find equation of a tangent
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

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