Systems of Linear Equations and Their Solution Sets · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

One point, no point, or a whole line

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

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

    • Consistent with a unique solution
    • Inconsistent
    • Dependent with infinite solutions
    • Not a linear system

  2. Solve: x + y = 7 and x - y = 3. What is x?

    Answer: ______________

  3. Which of these systems is inconsistent?

    • x + y = 1 together with x - y = 0
    • 2x = 4 together with y = 1
    • x = 0 together with y = 0
    • x + y = 1 together with x + y = 2

  4. Solve x plus y equals 7 and x minus y equals 3. What is y?

    Answer: ______________

  5. Why can two equations in three unknowns never have exactly one solution?

    • Adding a third unknown always makes the system inconsistent
    • One variable always stays free, so the answer is none or infinitely many
    • Two equations can only draw parallel lines

  6. Sam says x plus y equals 4 and 2 x plus 2 y equals 8 share every point with y equals 4 minus x. Is Sam right?

    Circle one:   True   False

  7. Sam says x + y = 4 and 2x + 2y = 8 have infinitely many solutions given by y = 4 - x. Is Sam right?

    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. Elimination ends with the line 0 equals 5. What does it mean?

    • No solution: the steps ask parallel lines to meet
    • Infinitely many solutions on one shared line
    • Exactly one solution hiding in the last row

  10. Solve 2 x plus y equals 10 and x minus y equals 2. 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. Consistent with a unique solution · One crossing point is one shared solution.
  2. 5 · Add the equations: 2x = 10, so x = 5.
  3. x + y = 1 together with x + y = 2 · The pair x + y = 1 together with x + y = 2 demands the same sum be both 1 and 2, impossible.
  4. 2 · With x equals 5, y is 7 minus 5, which is 2.
  5. One variable always stays free, so the answer is none or infinitely many · A free variable means zero solutions or a whole family, never a lone point.
  6. True · The second equation is double the first, so both draw the same line.
  7. True · Sam is right: the second equation is double the first, so every point with y = 4 - x works.
  8. Consistent with a unique solution · One crossing point is one shared solution.
  9. No solution: the steps ask parallel lines to meet · A contradiction means the lines are parallel, so the system is inconsistent.
  10. 4 · Adding kills y: 3 x is 12, so x is 4.
Worksheet · LightMySky