Math 417: Practice Problems for Section 1.1

These are additional practice problems for Section 1.1 in addition to Exercises 1-18 on page 5.
Find all solutions for each of the following linear systems of equations in three unknowns.

Systems with a unique solution

  1. 10x + 9y - z = 42
    18x + 12y - z = 80
    -x - 6y + z = 1

    Solution: (5, -2/3, 2)
  2. x - z = -1
    6x + y -z =12
    5x + y + z = 15

    Solution: (1, 8, 2)
  3. -x + 3y - z = -4
    x + 4y - z = 2
    4x - 2y + z = 13

    Solution: (3, 0 ,1)
  4. -x - 6y - z = 2
    5x - 5y - z = 13
    8x + 7y + z = 11

    Solution: (2, -1, 2)
  5. 6x + 2y - z = 4
    10x + 3y -z = 7
    -x - y + z = -1

    Solution: (0, 3, 2)
  6. 2x + 6y + z = 41
    3x + 5y + z = 36
    2x + 7y + z = 47

    Solution: (1, 6, 3)
  7. -3x - 14y - 3z = -44
    6x - 12y -3z = -21
    15x + 16y +3z = 73

    Solution: (2, 5/2, 1)
  8. 3x - y - z = 2
    10x - z = 17
    5x + 2y +z = 15

    Solution: (1, 8, -7)
  9. 3x - y - z = 2
    10x - z = 17
    5x + 2y +z = 15

    Solution: (1, 8, -7)
  10. 7x + 3y - z = 5
    12x + 4y - z = -5
    -3x - 2y + z = -6

    Solution: (-1, 5, 3)

 

Systems with no solutions

  1. 3x + y + 27z = 2
    5x + 2y + 47z = 6
    7x + y + 55z = -5
  2. 10x - 3y - 38z = 3
    7x - 2y - 26z = 2
    3x - 3y - 24z = 4
  3. -8x + 3y - 36z = -20
    -11x + 4y - 50z = 28
    -3x + 3y - 6z = 3
  4. -6x + 7y -50 = -33
    -7x + 8y - 58z = -38
    5x + 7y + 16z = -8

 

Systems with infinitely many solutions

  1. 5x +2y - 17z = 2
    7x + 3y -24z = 32
    x + 2y -5z = 14

    Solution: (2+3t, 6+t, t) for every real number t
  2. 25x - 4y + 50z = -63
    31x + 5y -62z = -78
    -x + 4y + 2z = 15

    Solution: (3+2t, 3, t) for every real number t
  3. 2x - y + 2z = 6
    2x + y + 3z = 2
    4x - 4y + 3z = 16

    Solution: (2-5t/4, -2-t/2, t) for every real number t
  4. 4x + y - z = -1
    3x - 2y + 2z = 2
    3y - 3z = -3

    Solution: (0, -1+t, t) for every real number t
  5. -3x - y + 3z = 1
    6x + 3y - 5z = -4
    3x + 6y + 2z = -11

    Solution: (0, -1+t, t) for every real number t