Worksheet: The General Solution of a System of Linear Equations

In this worksheet, we will practice finding the general solution of a system of linear equations whether it has a unique solution, an infinite number of solutions, or no solution.

Q1:

Find the general solution to the system of equations associated with the following augmented matrix: 1 0 2 1 1 2 0 1 0 1 2 1 1 2 0 0 1 3 1 0 1 0 2 2 .

  • A 𝑥 = 1 6 𝑡 , 𝑥 = 𝑡 , 𝑥 = 1 + 7 𝑡 , 𝑥 = 4 𝑡 , 𝑥 = 3 9 𝑡
  • B 𝑥 = 4 𝑡 , 𝑥 = 1 6 𝑡 , 𝑥 = 1 + 7 𝑡 , 𝑥 = 𝑡 , 𝑥 = 3 9 𝑡
  • C 𝑥 = 𝑡 , 𝑥 = 1 6 𝑡 , 𝑥 = 1 + 7 𝑡 , 𝑥 = 4 𝑡 , 𝑥 = 3 9 𝑡
  • D 𝑥 = 𝑡 , 𝑥 = 1 6 𝑡 , 𝑥 = 1 + 7 𝑡 , 𝑥 = 4 𝑡 , 𝑥 = 3 + 9 𝑡
  • E 𝑥 = 4 𝑡 , 𝑥 = 1 6 𝑡 , 𝑥 = 1 + 7 𝑡 , 𝑥 = 𝑡 , 𝑥 = 3 + 9 𝑡

Q2:

Find the general solution to the linear system 0 1 2 1 2 1 1 4 5 𝑥 𝑦 𝑧 = 1 1 1 .

  • A 3 1 0 + 𝑡 1 2 4
  • B 3 1 0 + 𝑡 3 2 1
  • C 1 1 1 + 𝑡 1 2 3
  • D 3 1 0 + 𝑡 1 2 3
  • E 1 1 1 + 𝑡 3 2 1

Q3:

Find the general solution to the linear system 1 0 1 1 1 1 1 0 3 1 3 2 3 3 0 3 𝑥 𝑦 𝑧 𝑤 = 1 2 4 3 .

  • A 1 2 4 3 + 𝑡 0 1 1 1
  • B 1 2 4 3 + 𝑡 1 1 1 0
  • C 2 1 1 0 + 𝑡 1 1 3 3
  • D 2 1 1 0 + 𝑡 1 1 1 0
  • E 2 1 1 0 + 𝑡 0 1 1 1

Q4:

Find the general solution to the linear system 1 1 2 1 2 0 3 4 4 𝑥 𝑦 𝑧 = 1 2 4 .

  • A 1 2 4 + 𝑡 4 2 1
  • B 0 1 0 + 𝑡 1 1 3
  • C 0 1 0 + 𝑡 4 2 1
  • D 0 1 0 + 𝑡 1 2 4
  • E 1 2 4 + 𝑡 1 2 4

Q5:

Find the general solution to the linear system 1 1 0 1 2 1 1 2 1 0 1 1 0 0 0 0 𝑥 𝑦 𝑧 𝑤 = 2 1 3 0 .

  • A 8 5 0 5 + 𝑡 1 1 1 0
  • B 8 5 0 5 + 𝑡 0 1 1 1
  • C 8 5 0 5 + 𝑡 1 1 1 0
  • D 2 1 3 0 + 𝑡 0 1 1 1
  • E 2 1 3 0 + 𝑡 1 1 1 0

Q6:

Find the general solution of the system with the augmented matrix 1 1 0 1 1 0 4 2 .

  • A 𝑧 = 𝑡 , 𝑦 = 4 𝑡 , 𝑥 = 4 𝑡
  • B 𝑧 = 4 𝑡 1 , 𝑦 = 4 𝑡 , 𝑥 = 2 𝑡
  • C 𝑧 = 4 𝑡 1 , 𝑦 = 𝑡 , 𝑥 = 2 + 4 𝑡
  • D 𝑧 = 𝑡 , 𝑦 = 4 𝑡 1 , 𝑥 = 2 4 𝑡
  • E 𝑧 = 𝑡 , 𝑦 = 4 𝑡 + 2 , 𝑥 = 2 3 𝑡

Q7:

Find the general solution of the system with the augmented matrix 1 2 0 2 2 0 1 1 3 2 1 3 .

  • A 𝑧 = 𝑡 , 𝑦 = 4 3 + 1 4 𝑡 , 𝑥 = 1 2 1 2 𝑡 , where 𝑡
  • B 𝑧 = 𝑡 , 𝑦 = 3 4 1 4 𝑡 , 𝑥 = 1 2 + 1 2 𝑡 , where 𝑡
  • C 𝑧 = 𝑡 , 𝑦 = 3 4 + 1 4 𝑡 , 𝑥 = 1 2 + 1 2 𝑡 , where 𝑡
  • D 𝑧 = 𝑡 , 𝑦 = 4 3 1 4 𝑡 , 𝑥 = 1 2 1 2 𝑡 , where 𝑡
  • E 𝑧 = 𝑡 , 𝑦 = 3 4 + 1 4 𝑡 , 𝑥 = 1 2 1 2 𝑡 , where 𝑡

Q8:

Which matrix below would you row-reduce to solve the following system of equations: 3 𝑥 2 𝑦 + 𝑧 = 3 , 2 𝑥 + 𝑦 𝑧 = 1 ?

  • A 3 2 1 3 2 1 1 1
  • B 2 3 1 2 1 1 1 3
  • C 3 2 2 1 1 1 3 1
  • D 3 2 1 2 1 1

Q9:

Which of the following augmented matrices represents the pair of equations 2 𝑥 3 𝑦 = 1 and 2 𝑥 3 𝑦 = 1 ?

  • A 2 2 3 3 1 1
  • B 2 2 3 3 1 1
  • C 2 2 3 3 1 1
  • D 2 3 1 2 3 1
  • E 2 3 1 2 3 1

Q10:

Find the general solution to the linear system 0 1 2 1 0 1 1 2 5 𝑥 𝑦 𝑧 = 1 1 1 .

  • A 1 1 0 + 𝑡 1 2 1
  • B 1 1 1 + 𝑡 1 2 1
  • C 1 1 1 + 𝑡 1 2 1
  • D 1 1 0 + 𝑡 2 1 5
  • E 1 1 0 + 𝑡 1 2 1

Q11:

Find the general solution to the linear system 1 1 0 1 1 1 1 0 3 1 1 2 3 3 0 3 𝑥 𝑦 𝑧 𝑤 = 1 2 4 3 .

  • A 𝑡 1 2 𝑠 1 2 𝑡 𝑠 1 2 𝑠 1 2 𝑡 + 1 2 4 3
  • B 𝑡 1 2 𝑠 1 2 𝑡 𝑠 1 2 𝑠 1 2 𝑡 + 3 2 1 2 0 0
  • C 1 2 𝑠 1 2 𝑡 1 2 𝑠 1 2 𝑡 𝑠 𝑡 + 1 2 4 3
  • D 𝑠 𝑡 𝑠 𝑡 𝑠 𝑡 + 3 2 1 2 0 0
  • E 1 2 𝑠 1 2 𝑡 1 2 𝑠 1 2 𝑡 𝑠 𝑡 + 3 2 1 2 0 0

Q12:

Find the general solution to the linear system 1 1 2 1 2 1 3 4 5 𝑥 𝑦 𝑧 = 1 2 4 .

  • A 0 1 0 + 𝑡 4 2 3
  • B 0 1 0 + 𝑡 3 1 1
  • C 1 2 4 + 𝑡 3 1 1
  • D 0 1 0 + 𝑡 1 1 3
  • E 1 2 4 + 𝑡 1 1 3

Q13:

Find the general solution to the linear system 1 1 0 1 2 1 1 2 1 0 1 1 0 1 1 1 𝑥 𝑦 𝑧 𝑤 = 2 1 3 1 .

  • A 2 1 3 1 + 𝑡 0 1 1 1
  • B 2 1 3 1 + 𝑡 1 1 1 0
  • C 9 5 0 6 + 𝑡 1 2 1 0
  • D 9 5 0 6 + 𝑡 1 1 1 0
  • E 9 5 0 6 + 𝑡 0 1 1 1

Q14:

Find the general solution to the linear system 1 2 3 2 1 0 2 1 1 2 1 4 4 3 3 0 2 1 1 2 𝑥 𝑥 𝑥 𝑥 𝑥 = 1 1 7 1 8 7 .

  • A 𝑤 𝑣 2 𝑢 1 2 𝑢 1 2 𝑣 𝑤 𝑢 𝑣 𝑤 + 1 1 7 1 8 7 0
  • B 1 2 𝑢 1 2 𝑣 𝑤 𝑤 𝑣 2 𝑢 𝑢 𝑣 𝑤 + 4 7 2 0 0 0
  • C 𝑤 𝑣 2 𝑢 1 2 𝑢 1 2 𝑣 𝑤 𝑢 𝑣 𝑤 + 4 7 2 0 0 0
  • D 𝑢 𝑣 𝑤 𝑤 𝑣 2 𝑢 1 2 𝑢 1 2 𝑣 𝑤 + 4 7 2 0 0 0
  • E 𝑢 𝑣 𝑤 𝑤 𝑣 2 𝑢 1 2 𝑢 1 2 𝑣 𝑤 + 1 1 7 1 8 7 0

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