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DSE M2 Systems of Linear Equations Practice

DSE M2 · 60 practice questions with instant marking and worked solutions.

Systems of Linear Equations is an important part of the DSE M2 mathematics curriculum. This topic develops Cramer's rule, systems of three equations, and deciding between a unique solution, no solution and infinitely many. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Systems of Linear Equations has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 60 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Systems of Linear Equations; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 12 of them are DSE Extended Part style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Systems of Linear Equations set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Systems of Linear Equations is closely linked with the rest of the Extended Part: algebraic and trigonometric skills underpin calculus, and calculus in turn supports probability distributions and statistical inference. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    Use Cramer's rule to solve $\begin{cases}3x+2y=10\\3x-y=4\end{cases}$. Find the value of $y$. Choose the correct answer.
  2. 2 Basic
    The system $\begin{cases}x+2y=1\\2x+ky=5\end{cases}$ has no unique solution. Find the value of $k$. Give the answer only.

  3. 3 Basic
    Consider the system $\begin{cases}3x+2y+2z=6\\-x+2y+2z=-10\\3x+z=11\end{cases}$. Find the value of $y$. Show the essential working and give the answer.

  4. 4 Basic
    The system $\begin{cases}2x+3y+3z=5\\-x+y+z=1\\5y+5z=c\end{cases}$ has a zero coefficient determinant. The value of $c$ for which the system has infinitely many solutions is 3. Is this true?
  5. 5 Basic
    Use Cramer's rule to solve $\begin{cases}3x-y=-5\\3x-3y=3\end{cases}$. Find the value of $y$. Choose the correct answer.
  6. 6 Basic
    The system $\begin{cases}2x+3y=1\\2x+ky=5\end{cases}$ has no unique solution. Find the value of $k$. Give the answer only.

  7. 7 Basic
    Consider the system $\begin{cases}3x+2y-2z=11\\3x+y-2z=10\\2y+z=2\end{cases}$. Find the value of $x$. Show the essential working and give the answer.

  8. 8 Basic
    The system $\begin{cases}-2x-y+2z=3\\-x-y+z=-4\\-4x-3y+4z=c\end{cases}$ has a zero coefficient determinant. The value of $c$ for which the system has infinitely many solutions is 11. Is this true?
  9. 9 Basic
    Use Cramer's rule to solve $\begin{cases}x-y=-3\\2x-y=-6\end{cases}$. Find the value of $y$. Choose the correct answer.
  10. 10 Basic
    The system $\begin{cases}4x+5y=1\\6x+ky=5\end{cases}$ has no unique solution. Find the value of $k$ (as a fraction in lowest terms). Give the answer only.

  11. 11 Basic
    Consider the system $\begin{cases}-x+2y+z=-3\\-x+3y-2z=-18\\3x-2y=9\end{cases}$. Find the value of $y$. Show the essential working and give the answer.

  12. 12 Basic
    The system $\begin{cases}-x=5\\3x+3y+z=4\\-4x-3y-z=c\end{cases}$ has a zero coefficient determinant. The value of $c$ for which the system has infinitely many solutions is 9. Is this true?
  13. 13 Standard
    The system $\begin{cases}2x-2y+2z=1\\-2x+y=2\\3y+kz=3\end{cases}$ does not have a unique solution. Find the value of $k$. Choose the correct answer.
  14. 14 Standard
    Use Cramer's rule to solve $\begin{cases}3x+2y=15\\3x-3y=0\end{cases}$. Find the value of $x$. Give the answer only.

  15. 15 Standard
    The system $\begin{cases}x+3y=1\\2x+ky=5\end{cases}$ has no unique solution. Find the value of $k$. Show the essential working and give the answer.

  16. 16 Standard DSE long question
    Consider the system $\begin{cases}2x+y=3\\-x+3y=-12\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
  17. 17 Standard
    The system $\begin{cases}-2x+3y-z=1\\-x+3y=2\\3x-2y+kz=3\end{cases}$ does not have a unique solution. Find the value of $k$ (as a fraction in lowest terms). Choose the correct answer.
  18. 18 Standard
    Use Cramer's rule to solve $\begin{cases}2x+2y=4\\-x+y=-2\end{cases}$. Find the value of $y$. Give the answer only.

  19. 19 Standard
    The system $\begin{cases}2x+5y=1\\4x+ky=5\end{cases}$ has no unique solution. Find the value of $k$. Show the essential working and give the answer.

  20. 20 Standard DSE long question
    Consider the system $\begin{cases}3x+y=2\\x+y=-2\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
  21. 21 Standard
    The system $\begin{cases}x-2y-z=1\\-x-2y+z=2\\-2x-2y+kz=3\end{cases}$ does not have a unique solution. Find the value of $k$. Choose the correct answer.
  22. 22 Standard
    Use Cramer's rule to solve $\begin{cases}2x-y=-1\\2x-3y=-3\end{cases}$. Find the value of $y$. Give the answer only.

  23. 23 Standard
    The system $\begin{cases}2x+2y=1\\4x+ky=5\end{cases}$ has no unique solution. Find the value of $k$. Show the essential working and give the answer.

  24. 24 Standard DSE long question
    Consider the system $\begin{cases}3x+3y=-3\\-x-2y=-1\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
  25. 25 Advanced
    Consider the system $\begin{cases}x-2z=0\\-x+3y+3z=-4\\x-y-z=0\end{cases}$. Find the value of $x$. Choose the correct answer.
  26. 26 Advanced
    The system $\begin{cases}-2x+y+3z=0\\x-2y+2z=-4\\-3x+3y+z=c\end{cases}$ has a zero coefficient determinant. Find the value of $c$ for which the system has infinitely many solutions. Give the answer only.

  27. 27 Advanced
    The system $\begin{cases}-x+2y+z=1\\-x+3z=2\\-x-2y+kz=3\end{cases}$ does not have a unique solution. The value of $k$ is 5. Is this true?
  28. 28 Advanced DSE long question
    Consider the system $\begin{cases}x+y+z=-3\\-2x+2y-z=5\\3x+2y-2z=-4\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
    4. (d) Find $z$.
  29. 29 Advanced
    Consider the system $\begin{cases}x+2y+2z=4\\-x+y-2z=-1\\3x-y-2z=13\end{cases}$. Find the value of $z$. Choose the correct answer.
  30. 30 Advanced
    The system $\begin{cases}3x+2y+3z=-1\\3x-2y+2z=6\\9x-2y+7z=c\end{cases}$ has a zero coefficient determinant. Find the value of $c$ for which the system has infinitely many solutions. Give the answer only.

  31. 31 Advanced
    The system $\begin{cases}3x-y+3z=1\\-2x-2y-z=2\\3x-2y+kz=3\end{cases}$ does not have a unique solution. The value of $k$ is 27/8. Is this true?
  32. 32 Advanced DSE long question
    Consider the system $\begin{cases}x+y=2\\y+z=1\\y=4\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
    4. (d) Find $z$.
  33. 33 Advanced
    Consider the system $\begin{cases}-2x-z=8\\-x+2y+z=3\\2x-2y-z=-6\end{cases}$. Find the value of $y$. Choose the correct answer.
  34. 34 Advanced
    The system $\begin{cases}x-2y+3z=-4\\3y+z=5\\x+y+4z=c\end{cases}$ has a zero coefficient determinant. Find the value of $c$ for which the system has infinitely many solutions. Give the answer only.

  35. 35 Advanced
    The system $\begin{cases}3x-2y=1\\-x+3y=2\\-x-y+kz=3\end{cases}$ does not have a unique solution. The value of $k$ is 0. Is this true?
  36. 36 Advanced DSE long question
    Consider the system $\begin{cases}-x+y+z=-4\\-2x+2z=-6\\z=-3\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
    4. (d) Find $z$.
  37. 37 Challenging
    Use Cramer's rule to solve $\begin{cases}2x+y=-2\\-x-3y=1\end{cases}$. Find the value of $x$. Choose the correct answer.
  38. 38 Challenging
    The system $\begin{cases}3x+2y=1\\3x+ky=5\end{cases}$ has no unique solution. Find the value of $k$. Show the essential working and give the answer.

  39. 39 Challenging
    Consider the system $\begin{cases}-x-y+2z=-4\\x+3y+3z=22\\2y+2z=12\end{cases}$. The value of $y$ is 2. Is this true?
  40. 40 Challenging DSE long question
    Consider the system $\begin{cases}2x+3y=-16\\-x-2y=10\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
  41. 41 Challenging
    Use Cramer's rule to solve $\begin{cases}3x-y=-21\\-x+y=11\end{cases}$. Find the value of $x$. Choose the correct answer.
  42. 42 Challenging
    The system $\begin{cases}3x+3y=1\\6x+ky=5\end{cases}$ has no unique solution. Find the value of $k$. Show the essential working and give the answer.

  43. 43 Challenging
    Consider the system $\begin{cases}2y-z=-7\\3x-2y+z=10\\-2x+2y=-6\end{cases}$. The value of $x$ is -2. Is this true?
  44. 44 Challenging DSE long question
    Consider the system $\begin{cases}3x+y=3\\2x-2y=10\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
  45. 45 Challenging
    Use Cramer's rule to solve $\begin{cases}x-y=9\\3x-y=17\end{cases}$. Find the value of $x$. Choose the correct answer.
  46. 46 Challenging
    The system $\begin{cases}3x+2y=1\\4x+ky=5\end{cases}$ has no unique solution. Find the value of $k$ (as a fraction in lowest terms). Show the essential working and give the answer.

  47. 47 Challenging
    Consider the system $\begin{cases}x+3y-z=7\\3x+y-2z=4\\-2x-y+z=-3\end{cases}$. The value of $z$ is 0. Is this true?
  48. 48 Challenging DSE long question
    Consider the system $\begin{cases}2x+3y=-3\\x+3y=0\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
  49. 49 Elite
    The system $\begin{cases}2x-y=-3\\3x+y-2z=4\\-x-2y+2z=c\end{cases}$ has a zero coefficient determinant. Find the value of $c$ for which the system has infinitely many solutions. Give the answer only.

  50. 50 Elite
    The system $\begin{cases}x-2y+2z=1\\-x-y-z=2\\2x+kz=3\end{cases}$ does not have a unique solution. Find the value of $k$ (as a fraction in lowest terms). Show the essential working and give the answer.

  51. 51 Elite
    Use Cramer's rule to solve $\begin{cases}x+2y=12\\3x-y=15\end{cases}$. The value of $x$ is 6. Is this true?
  52. 52 Elite DSE long question
    Consider the system $\begin{cases}2x+y-2z=0\\3x+2y=5\\3x-z=7\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
    4. (d) Find $z$.
  53. 53 Elite
    The system $\begin{cases}2x+3y+3z=1\\-2z=1\\2x+3y+z=c\end{cases}$ has a zero coefficient determinant. Find the value of $c$ for which the system has infinitely many solutions. Give the answer only.

  54. 54 Elite
    The system $\begin{cases}3x+2y+3z=1\\-x+2z=2\\2x+kz=3\end{cases}$ does not have a unique solution. Find the value of $k$. Show the essential working and give the answer.

  55. 55 Elite
    Use Cramer's rule to solve $\begin{cases}x+2y=1\\3x+y=13\end{cases}$. The value of $x$ is 5. Is this true?
  56. 56 Elite DSE long question
    Consider the system $\begin{cases}2x-z=2\\3x-y-2z=1\\-2x+2y+3z=0\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
    4. (d) Find $z$.
  57. 57 Elite
    The system $\begin{cases}-x+2y+2z=0\\3x+3y+3z=3\\5x+8y+8z=c\end{cases}$ has a zero coefficient determinant. Find the value of $c$ for which the system has infinitely many solutions. Give the answer only.

  58. 58 Elite
    The system $\begin{cases}2x+2y-2z=1\\-x-2y+3z=2\\-2x+2y+kz=3\end{cases}$ does not have a unique solution. Find the value of $k$. Show the essential working and give the answer.

  59. 59 Elite
    Use Cramer's rule to solve $\begin{cases}x+2y=-7\\-x+y=-8\end{cases}$. The value of $x$ is 3. Is this true?
  60. 60 Elite DSE long question
    Consider the system $\begin{cases}3x+y+3z=3\\x-2y-z=6\\-x+2y-z=-8\end{cases}$. Answer each part; give the answer only.
    1. (a) Find the coefficient determinant $\Delta$.
    2. (b) Find $x$.
    3. (c) Find $y$.
    4. (d) Find $z$.

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Systems of Linear Equations

This topic needs particular care with Cramer's rule, systems of three equations, and deciding between a unique solution, no solution and infinitely many. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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DSE M2 mixed revision → My mistakes →

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Repeating the same mistake usually means an earlier concept never landed. We find that step in the trial lesson.

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