Gaussian Elimination and Row Echelon Form · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Staircases that solve systems

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

  1. Solve x + y = 5 and x minus y = 1. What is x?

    Answer: ______________

  2. Solve x plus y equals 5 and x minus y equals 1. What is x?

    Answer: ______________

  3. A staircase row reading 0 equals 5 means the system has no solution.

    Circle one:   True   False

  4. Which list gives the three allowed row operations?

    • Swap columns, delete a row, multiply every solution by 2
    • Swap rows, scale a row by a nonzero number, add a multiple of one row to another
    • Add the columns, scale a row by zero, swap the two sides

  5. The reduced form of a 3 by 4 system has pivots in columns 1 and 2. Which columns are free?

    • Columns 3 and 4
    • Columns 1 and 2
    • Column 1 only
    • No free columns

  6. A system has 5 unknowns and its staircase form has 3 pivots. How many free variables are there?

    Answer: ______________

  7. Solve x plus y equals 7 and 2 x minus y equals 5. What is x?

    • 3
    • 5
    • 4

  8. Solve x + y = 7 and 2x minus y = 5. What is x?

    • 4
    • 3
    • 5
    • 2

  9. Solve x plus y plus z equals 6, x plus y minus z equals 0, and x minus y equals negative 1. What is z?

    Answer: ______________

  10. A system has 5 unknowns and its echelon form has 3 pivots. How many free variables are there?

    Answer: ______________

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Answer key

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

Staircases that solve systems W1-mt_PZBce3UvRm-s1

  1. 3 · Adding gives 2x = 6, so x = 3.
  2. 3 · Adding kills y: 2 x is 6, so x is 3.
  3. True · No values can satisfy it, so the system is inconsistent.
  4. Swap rows, scale a row by a nonzero number, add a multiple of one row to another · Swaps, nonzero scalings, and adding multiples are all reversible, so solutions stay put.
  5. Columns 3 and 4 · Pivot columns are 1 and 2, so the remaining columns 3 and 4 are free.
  6. 2 · Free variables are unknowns minus pivots: 5 minus 3 is 2.
  7. 4 · Adding gives 3 x equals 12, so x is 4 and y is 3.
  8. 4 · Add the equations: 3x = 12 gives x = 4, then y = 3.
  9. 3 · Adding the first two gives x plus y equals 3, so z is 6 minus 3.
  10. 2 · Free variables equal unknowns minus rank: 5 minus 3 = 2.
Worksheet · LightMySky