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: 102112010121120013101022.

  • A𝑥=4𝑡, 𝑥=16𝑡, 𝑥=1+7𝑡, 𝑥=𝑡, 𝑥=3+9𝑡
  • B𝑥=4𝑡, 𝑥=16𝑡, 𝑥=1+7𝑡, 𝑥=𝑡, 𝑥=39𝑡
  • C𝑥=𝑡, 𝑥=16𝑡, 𝑥=1+7𝑡, 𝑥=4𝑡, 𝑥=39𝑡
  • D𝑥=16𝑡, 𝑥=𝑡, 𝑥=1+7𝑡, 𝑥=4𝑡, 𝑥=39𝑡
  • E𝑥=𝑡, 𝑥=16𝑡, 𝑥=1+7𝑡, 𝑥=4𝑡, 𝑥=3+9𝑡

Q2:

Find the general solution to the linear system 012121145𝑥𝑦𝑧=111.

  • A310+𝑡124
  • B310+𝑡123
  • C310+𝑡321
  • D111+𝑡321
  • E111+𝑡123

Q3:

Find the general solution to the linear system 1011111031323303𝑥𝑦𝑧𝑤=1243.

  • A1243+𝑡1110
  • B2110+𝑡0111
  • C2110+𝑡1110
  • D2110+𝑡1133
  • E1243+𝑡0111

Q4:

Find the general solution to the linear system 112120344𝑥𝑦𝑧=124.

  • A010+𝑡124
  • B124+𝑡421
  • C124+𝑡124
  • D010+𝑡113
  • E010+𝑡421

Q5:

Find the general solution to the linear system 1101211210110000𝑥𝑦𝑧𝑤=2130.

  • A8505+𝑡1110
  • B8505+𝑡0111
  • C8505+𝑡1110
  • D2130+𝑡1110
  • E2130+𝑡0111

Q6:

Find the general solution of the system with the augmented matrix 11011042.

  • A𝑧=4𝑡1, 𝑦=4𝑡, 𝑥=2𝑡
  • B𝑧=𝑡, 𝑦=4𝑡+2, 𝑥=23𝑡
  • C𝑧=4𝑡1, 𝑦=𝑡, 𝑥=2+4𝑡
  • D𝑧=𝑡, 𝑦=4𝑡1, 𝑥=24𝑡
  • E𝑧=𝑡, 𝑦=4𝑡, 𝑥=4𝑡

Q7:

Find the general solution of the system with the augmented matrix 120220113213.

  • A𝑧=𝑡, 𝑦=34+14𝑡, 𝑥=1212𝑡, where 𝑡
  • B𝑧=𝑡, 𝑦=4314𝑡, 𝑥=1212𝑡, where 𝑡
  • C𝑧=𝑡, 𝑦=34+14𝑡, 𝑥=12+12𝑡, where 𝑡
  • D𝑧=𝑡, 𝑦=43+14𝑡, 𝑥=1212𝑡, where 𝑡
  • E𝑧=𝑡, 𝑦=3414𝑡, 𝑥=12+12𝑡, where 𝑡

Q8:

Which matrix below would you row-reduce to solve the following system of equations: 3𝑥2𝑦+𝑧=3,2𝑥+𝑦𝑧=1?

  • A23121113
  • B32132111
  • C32211131
  • D321211

Q9:

Find the general solution to the linear system 012101125𝑥𝑦𝑧=111.

  • A111+𝑡121
  • B110+𝑡121
  • C111+𝑡121
  • D110+𝑡121
  • E110+𝑡215

Q10:

Find the general solution to the linear system 1101111031123303𝑥𝑦𝑧𝑤=1243.

  • A𝑠𝑡𝑠𝑡𝑠𝑡+321200
  • B𝑡12𝑠12𝑡𝑠12𝑠12𝑡+321200
  • C12𝑠12𝑡12𝑠12𝑡𝑠𝑡+1243
  • D12𝑠12𝑡12𝑠12𝑡𝑠𝑡+321200
  • E𝑡12𝑠12𝑡𝑠12𝑠12𝑡+1243

Q11:

Find the general solution to the linear system 112121345𝑥𝑦𝑧=124.

  • A010+𝑡423
  • B124+𝑡311
  • C010+𝑡113
  • D010+𝑡311
  • E124+𝑡113

Q12:

Find the general solution to the linear system 1101211210110111𝑥𝑦𝑧𝑤=2131.

  • A9506+𝑡1210
  • B2131+𝑡1110
  • C9506+𝑡0111
  • D2131+𝑡0111
  • E9506+𝑡1110

Q13:

Find the general solution to the linear system 12321021121443302112𝑥𝑥𝑥𝑥𝑥=117187.

  • A12𝑢12𝑣𝑤𝑤𝑣2𝑢𝑢𝑣𝑤+472000
  • B𝑤𝑣2𝑢12𝑢12𝑣𝑤𝑢𝑣𝑤+1171870
  • C𝑢𝑣𝑤𝑤𝑣2𝑢12𝑢12𝑣𝑤+472000
  • D𝑢𝑣𝑤𝑤𝑣2𝑢12𝑢12𝑣𝑤+1171870
  • E𝑤𝑣2𝑢12𝑢12𝑣𝑤𝑢𝑣𝑤+472000

Q14:

Consider the matrix 𝐴=106159.

Find the matrix with the smallest positive integer entries, 𝑋,such that 𝐴𝑋=00.

  • A𝑋=106
  • B𝑋=35
  • C𝑋=915
  • D𝑋=53
  • E𝑋=610

Find the matrix with the smallest non-zero positive integer entries, 𝑇, such that 𝐴𝑇=1015.

  • A𝑇=54
  • B𝑇=13
  • C𝑇=31
  • D𝑇=43
  • E𝑇=45

Q15:

Find the solution of the system of equations 7𝑥+14𝑦+15𝑧=22, 2𝑥+4𝑦+3𝑧=5, 3𝑥+6𝑦+10𝑧=13, giving your answer in terms of an arbitrary real number 𝑡 if necessary.

  • A𝑥=12𝑡, 𝑧=1, 𝑦=𝑡
  • B𝑥=13𝑡, 𝑧=1, 𝑦=2𝑡
  • C𝑥=1+2𝑡, 𝑧=1, 𝑦=𝑡
  • D𝑥=1+2𝑡, 𝑧=1, 𝑦=𝑡
  • E𝑥=12𝑡, 𝑧=1, 𝑦=𝑡

Q16:

Find the solution of the system of equations 3𝑥𝑦+4𝑧=6, 𝑦+8𝑧=0, 2𝑥+𝑦=4, giving your answer in terms of an arbitrary real number 𝑡 if necessary.

  • A𝑥=2+4𝑡, 𝑧=𝑡, 𝑦=8𝑡
  • B𝑥=2+4𝑡, 𝑧=𝑡, 𝑦=8𝑡
  • C𝑥=24𝑡, 𝑧=𝑡, 𝑦=8𝑡
  • D𝑥=24𝑡, 𝑧=𝑡, 𝑦=8𝑡
  • E𝑥=2+3𝑡, 𝑧=2𝑡, 𝑦=8𝑡

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