Systems of linear equations and their solution sets · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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One point, no point, or a whole line

Mathematics · Linear Algebra · ages 18-19
Name ______________________   Date ____________

  1. What does a free parameter do in an infinite solution set?

    • It names different solutions in the family
    • It forces the parameter to equal zero
    • It proves that the system has no solution

  2. In a system of two equations in two variables, the two lines are parallel and distinct. How many solutions does the system have?

    Answer: ______________

  3. Which elimination result shows dependent equations with infinitely many solutions in a two-equation, two-variable system?

    • 0 = 0
    • 0 = 5
    • x = 4

  4. Solve 2x + y = 11 and x - y = 1. What is y?

    Answer: ______________

  5. Why can two linear equations in three unknowns not have exactly one solution?

    • An inconsistent system has no solution, while a consistent one has a free variable
    • Every third unknown makes the two equations inconsistent
    • Two equations in three unknowns always describe parallel lines

  6. Sam says x + y = 4 and 2x + 2y = 8 share every point with y = 4 - x. Is Sam right?

    Circle one:   True   False

  7. Two distinct, nonparallel lines in a two-variable system give exactly one solution.

    Circle one:   True   False

  8. Two lines cross at exactly one point. How is the system classified?

    • Inconsistent
    • Dependent with infinite solutions
    • Consistent with a unique solution

  9. For a two-variable system, elimination ends with 0 = 5. What does this mean?

    • No solution because the two lines cannot share a point
    • Infinitely many solutions because the two lines are the same
    • One solution because a point is hidden in the last row

  10. Solve x + 2y = 8 and 3x - y = 3 by elimination. What is x?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

One point, no point, or a whole line W1-mt_sQL6JfR7Te-s1

  1. It names different solutions in the family · A free parameter can take allowed values, and each value gives a different member of the solution family.
  2. 0 · Parallel distinct lines never meet, so no point lies on both lines. No pair satisfies both equations, so the system has 0 solutions and is inconsistent.
  3. 0 = 0 · The identity 0 = 0 is consistent. With two equations in two variables it shows that the equations describe the same line, giving infinitely many solutions.
  4. 3 · Add the equations to get 3x = 12, so x = 4. Then 4 - y = 1, giving y = 3.
  5. An inconsistent system has no solution, while a consistent one has a free variable · If the system is inconsistent, it has no solution. If it is consistent, at least one variable remains free, giving infinitely many solutions. Neither case gives one isolated solution.
  6. True · The second equation is double the first, so both draw the same line.
  7. True · Distinct lines that are not parallel must cross, and two distinct straight lines share at most one point. That single shared point satisfies both equations.
  8. Consistent with a unique solution · One crossing point is one shared solution.
  9. No solution because the two lines cannot share a point · A contradiction means the lines are parallel, so the system is inconsistent.
  10. 2 · Double the second equation to get 6x - 2y = 6. Adding the first equation gives 7x = 14, so x = 2. Substitution gives y = 3.
Worksheet · LightMySky