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IGCSE · 4MA1 · Topic

Solving Linear Equations: past paper questions & worked solutions

Every Solving Linear Equations question from the 4MA1 papers in our bank, 2020–2026, sorted from easiest to hardest. Try each one first, then open the worked solution.

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14questions
87marks in total
26%of papers (14 of 54) include it
0/0/14easy / medium / hard

Questions, easiest first

Level is estimated from marks: 1–2 easy, 3–4 medium, 5+ hard.

1. Jun 2025 P2HR · Q1

Hard5 marks
Jun 2025 P2HR Question 1 (5 marks): Solving Linear Equations – 1 (a) Factorise fully 18 45 c cd - (2) (b) Solve 5 2 6 3 4 x x Show clear algebraic working. x = (3) (Total for Question 1 is 5 marks)
Show worked solution
  1. (a)
    \[ 18c-45cd=9c(2-5d) \]
  2. (b)
    \[ \frac{5-2x}{6}=3x-4 \]
    \[ 5-2x=18x-24 \]
    \[ 29=20x \]
    \[ x=\frac{29}{20} \]
Final answer

(a) \(9c(2-5d)\), (b) \(x=\frac{29}{20}\)

2. May 2024 P1H · Q6

Hard5 marks
May 2024 P1H Question 6 (5 marks): Solving Linear Equations – 6 (a) Expand and simplify (m + 5) (m - 8) (2) (b) Solve 3n - 4 = 5 6 3 n + Show clear algebraic working. n = (3) (Total for Question 6 is 5 marks)
Show worked solution
  1. (a)
    \[ (m+5)(m-8)=m^2-3m-40 \]
  2. (b)
    \[ 3n-4=\frac{5n+6}{3} \]
    \[ 9n-12=5n+6 \]
    \[ 4n=18 \]
Final answer

(a) \(m^2-3m-40\), (b) \(n=\frac92\)

3. Jun 2022 P1H · Q5

Hard5 marks
Jun 2022 P1H Question 5 (5 marks): Solving Linear Equations – 5 (a) Expand and simplify (n - 6)(n + 4) (2) (b) Solve 2x - 3 = 3 5 4 x - Show clear algebraic working. x = (3) (Total for Question 5 is 5 marks)
Show worked solution
  1. (a)
    \[ (n-6)(n+4)=n^2-2n-24 \]
  2. (b)
    \[ 2x-3=\frac{3x-5}{4} \]
    \[ 8x-12=3x-5 \]
    \[ 5x=7 \]
Final answer

(a) \(n^2-2n-24\), (b) \(x=\frac75\)

4. Jan 2022 P2HR · Q3

Hard5 marks
Jan 2022 P2HR Question 3 (5 marks): Solving Linear Equations – 3 (a) Solve 4y + 5 > 12 (2) (b) Solve 6 5 4 7 2 x x Show clear algebraic working. x = (3) (Total for Question 3 is 5 marks)
Show worked solution
  1. (a)
    \[ 4y+5>12 \]
    \[ 4y>7 \]
    \[ y>\frac74 \]
  2. (b)
    \[ 6x-5=\frac{4x-7}{2} \]
  3. Multiply by 2

    Multiply by 2: \[ 12x-10=4x-7 \]

    \[ 8x=3 \]
    \[ x=\frac38 \]
Final answer

\(y>\frac74\), and \(x=\frac38\).

5. Jan 2021 P1H · Q5

Hard5 marks
Jan 2021 P1H Question 5 (5 marks): Solving Linear Equations – 5 (a) Factorise fully 15y 4 + 20uy 3 (2) (b) Solve 4 - 3x = 5 8 4 - x Show clear algebraic working. x = (3) (Total for Question 5 is 5 marks)
Show worked solution
  1. (a)
    \[ 15y^4+20uy^3=5y^3(3y+4u) \]
  2. (b)
    \[ 4-3x=\frac{5-8x}{4} \]
    \[ 16-12x=5-8x \]
    \[ 11=4x \]
Final answer

(a) \(5y^3(3y+4u)\), (b) \(x=\frac{11}{4}\)

6. Specimen 4WM1H · Q4

Hard5 marks
Specimen 4WM1H Question 4 (5 marks): Solving Linear Equations – 4 (a) Expand and simplify (n - 6)(n + 4). (b) Solve 2x - 3 = (3x - 5)/4. (Total for Question 4 is 5 marks)
Show worked solution
  1. Expand the brackets
    \[ (n-6)(n+4)=n^2+4n-6n-24 \]
    \[ =n^2-2n-24 \]
  2. Clear the denominator
    \[ 2x-3=\frac{3x-5}{4} \]
    \[ 4(2x-3)=3x-5 \]
  3. Solve the equation
    \[ 8x-12=3x-5 \]
    \[ 5x=7 \]
    \[ x=\frac75 \]
Final answer

\(n^2-2n-24\), and \(x=\frac75\).

7. Jun 2025 P1HR · Q15

Hard6 marks
Jun 2025 P1HR Question 15 (6 marks): Solving Linear Equations – 15 (a) Solve 5 8 3 2 5 4 23 a a Show clear algebraic working. a = (4) (b) Express y 3 1    in the form cyn where c and n are numbers to be found. (2) (Total for Question 15 is 6 marks)
Show worked solution
  1. Solve equation

    For part (a),

    \[ \frac{5a+8}{3}-\frac{2a+5}{4}=23 \]
  2. Multiply by

    Multiply by \(12\):

    \[ 4(5a+8)-3(2a+5)=276 \]
    \[ 20a+32-6a-15=276 \]
    \[ 14a+17=276 \]
    \[ 14a=259 \]
    \[ a=\frac{259}{14}=\frac{37}{2} \]
  3. Solve equation

    For part (b),

    \[ \left(\frac{\sqrt y}{3}\right)^{-1}=\frac{3}{\sqrt y} \]
    \[ =3y^{-1/2} \]
Final answer

(a) \(a=\frac{37}{2}\), (b) \(3y^{-1/2}\)

8. May 2025 4WM1HR · Q12

Hard6 marks
May 2025 4WM1HR Question 12 (6 marks): Solving Linear Equations – 12 (a) Solve 5 8 3 2 5 4 23 a a Show clear algebraic working. a = (4) (b) Express y 3 1    in the form cyn where c and n are numbers to be found. (2) (Total for Question 12 is 6 marks)
Show worked solution
  1. Solve equation

    For part (a),

    \[ \frac{5a+8}{3}-\frac{2a+5}{4}=23 \]
  2. Multiply by

    Multiply by \(12\):

    \[ 4(5a+8)-3(2a+5)=276 \]
    \[ 20a+32-6a-15=276 \]
    \[ 14a+17=276 \]
    \[ 14a=259 \]
    \[ a=\frac{259}{14}=\frac{37}{2} \]
  3. Solve equation

    For part (b),

    \[ \left(\frac{\sqrt y}{3}\right)^{-1}=\frac{3}{\sqrt y} \]
    \[ =3y^{-1/2} \]
Final answer

(a) \(a=\frac{37}{2}\), (b) \(3y^{-1/2}\)

9. Nov 2024 P1H · Q3

Hard6 marks
Nov 2024 P1H Question 3 (6 marks): Solving Linear Equations – 3 (a) Simplify p3 5 (1) (b) Expand and simplify 2 4 3 4 n n n n (2) (c) Solve 2 5 3 4 x x Show clear algebraic working. x = (3) (Total for Question 3 is 6 marks)
Show worked solution
  1. (a)

    (a) \((p^3)^5=p^{15}\).

  2. (b)
    \[ 2n(4n+3)+n(n-4)=8n^2+6n+n^2-4n=9n^2+2n \]
  3. (c)
    \[ \frac{2x+5}{3}=4-x \]
    \[ 2x+5=12-3x \]
    \[ 5x=7 \]
Final answer

(a) \(p^{15}\), (b) \(9n^2+2n\), (c) \(x=\frac75\)

10. May-Nov 2020 P1HR · Q3

Hard6 marks
May-Nov 2020 P1HR Question 3 (6 marks): Solving Linear Equations – 3 (a) Simplify h7× h2 (1) G = c2 - 4c (b) Find the value of G when c = -5 G = (2) (c) Solve 5 3 4 x - = 2x + 3 Show clear algebraic working. x = (3) (Total for Question 3 is 6 marks)
Show worked solution
  1. (a)
    \[ h^7\times h^2=h^{7+2}=h^9 \]
  2. (b)
    \[ G=c^2-4c \]
  3. Solve quadratic equation

    When \(c=-5\): \[ G=(-5)^2-4(-5)=25+20=45 \]

  4. (c)
    \[ \frac{5x-3}{4}=2x+3 \]
  5. Multiply by 4

    Multiply by 4: \[ 5x-3=8x+12 \]

    \[ -3x=15 \]
    \[ x=-5 \]
Final answer

\(h^9\), \(G=45\), and \(x=-5\).

11. Jun 2026 P2HR · Q4

Hard8 marks
Jun 2026 P2HR Question 4 (8 marks): Solving Linear Equations – 4 (a) Solve 3(x + 2) = 1 – 5x Show clear algebraic working. x = (3) P = c 2 – 5d (b) Work out the value of P when c = –7 and d = 3 P = (2) (c) Simplify m 0 (1) (d) Simplify (4r 2t) 3 (2) (Total for Question 4 is
Show worked solution
  1. Part (a) | Step 1
    \[3x+6=1-5x.\]

    Expand the bracket.

  2. Part (a) | Step 2
    \[8x=-5\quad\Rightarrow\quad x=-\frac58.\]

    Add 5x and subtract 6 on both sides, then divide by 8.

  3. Part (b) | Step 1
    \[P=(-7)^2-5(3)=49-15=34.\]

    The square applies to the whole negative number, so (-7) squared is positive.

  4. Part (c) | Step 1

    \(m^0=1\) for \(m\ne0\).

    Any nonzero base to the zero power equals 1.

  5. Part (d) | Step 1
    \[(4r^2t)^3=4^3(r^2)^3t^3=64r^6t^3.\]

    Cube each factor and multiply the exponents in the power of a power.

Final answer

(a) \(x=-\frac58\).

(b) \(P=34\).

(c) \(1\).

(d) \(64r^6t^3\).

12. Jun 2024 P2HR · Q6

Hard8 marks
Jun 2024 P2HR Question 6 (8 marks): Solving Linear Equations – 6 (a) Solve 2 3 4 17 f f Show clear algebraic working.  f = (3) (b) Simplify (e + 12)0 where e > 0  (1) (c) Simplify fully 12 4 4 6 2 a h ah  (2) (d) Factorise fully 20x5y + 12x3y4  (2) (Total for Question 6
Show worked solution
  1. (a)
    \[ \frac{2f}{3}=4f-17 \]
    \[ 2f=12f-51 \]
    \[ 10f=51 \]
    \[ f=\frac{51}{10} \]
  2. (b)

    (b) \((e+12)^0=1\).

  3. (c)
    \[ \frac{12a^4h^6}{4ah^2}=3a^3h^4 \]
  4. (d)
    \[ 20x^5y+12x^3y^4=4x^3y(5x^2+3y^3) \]
Final answer

\(f=\frac{51}{10}\), \(1\), \(3a^3h^4\), \(4x^3y(5x^2+3y^3)\)

13. Jun 2022 P2HR · Q6

Hard8 marks
Jun 2022 P2HR Question 6 (8 marks): Solving Linear Equations – 6 (a) Solve p = 3 5 10 p - Show clear algebraic working. p = (3) (b) Simplify a 0 where a > 0 (1) (c) Simplify fully 3 6 3 2 xy x y (2) (d) Factorise fully 10c 3d 2 + 15cd 4 (2) (Total for Question 6 is 8 marks)
Show worked solution
  1. (a)
    \[ p=\frac{3p-5}{10} \]
    \[ 10p=3p-5 \]
    \[ 7p=-5 \]
    \[ p=-\frac57 \]
  2. (b)

    (b) \(a^0=1\).

  3. (c)
    \[ \frac{3xy^3}{6x^2y}=\frac{y^2}{2x} \]
  4. (d)
    \[ 10c^3d^2+15cd^4=5cd^2(2c^2+3d^2) \]
Final answer

(a) \(p=-\frac57\), (b) \(1\), (c) \(\frac{y^2}{2x}\), (d) \(5cd^2(2c^2+3d^2)\)

14. Jan 2020 P1H · Q7

Hard9 marks
Jan 2020 P1H Question 7 (9 marks): Solving Linear Equations – 7 (a) Expand and simplify (m - 8)(m + 5) (2) (b) Factorise fully 5y + 20y2 (2) (c) Simplify ( p2 + 3)0 (1) (d) Solve 3(2x - 5) = 9 2 -x Show clear algebraic working. x = (4) (Total for Question 7 is 9 marks)
Show worked solution
  1. (a)

    (a) \((m-8)(m+5)=m^2-3m-40\).

  2. (b)

    (b) \(5y+20y^2=5y(1+4y)\).

  3. (c)

    (c) \((p^2+3)^0=1\).

  4. (d)
    \[ 3(2x-5)=\frac{9-x}{2} \]
    \[ 12x-30=9-x \]
    \[ 13x=39 \]
Final answer

(a) \(m^2-3m-40\), (b) \(5y(1+4y)\), (c) \(1\), (d) \(x=3\)

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