Worksheet: Systems of Linear Equations in Three Variables

In this worksheet, we will practice solving systems of linear equations in three variables that can have one solution, infinite solutions, or no solution.

Q1:

Solve the simultaneous equations 2π‘₯+3𝑦+2𝑧=215,βˆ’6π‘₯βˆ’2𝑦+7𝑧=βˆ’675,π‘₯+5𝑦+3𝑧=225.

  • Aπ‘₯=65, 𝑦=1, 𝑧=βˆ’35
  • Bπ‘₯=752315, 𝑦=187315, 𝑧=5189
  • Cπ‘₯=1,344505, 𝑦=221101, 𝑧=βˆ’195101
  • Dπ‘₯=1,478565, 𝑦=877565, 𝑧=βˆ’507565
  • Eπ‘₯=244255, 𝑦=18185, 𝑧=βˆ’21985

Q2:

The sum of the ages of three brothers is 123 years. The first brother is 3 years older than the second brother who is 9 years older than the third. Find their current ages.

  • A34 years, 37 years, 43 years
  • B37 years, 46 years, 49 years
  • C34 years, 37 years, 46 years
  • D34 years, 43 years, 46 years
  • E37 years, 40 years, 46 years

Q3:

Given that the solution set of the simultaneous equations βˆ’2π‘₯+9𝑦+2𝑧=π‘Ž,βˆ’4π‘₯+9π‘¦βˆ’3𝑧=𝑏,4π‘₯βˆ’3𝑦+8𝑧=𝑐 is {(βˆ’6,7,8)}, find the values of π‘Ž,𝑏, and 𝑐.

  • Aπ‘Ž=βˆ’35, 𝑏=βˆ’63, 𝑐=19
  • Bπ‘Ž=βˆ’21905, 𝑏=4647, 𝑐=βˆ’1,124683
  • Cπ‘Ž=βˆ’6, 𝑏=7, 𝑐=8
  • Dπ‘Ž=91, 𝑏=63, 𝑐=19
  • Eπ‘Ž=14687, 𝑏=4967, 𝑐=20953

Q4:

Given that the solution set of the simultaneous equations βˆ’7π‘₯+7π‘¦βˆ’6𝑧=24,8π‘₯+3π‘¦βˆ’4𝑧=6,8π‘₯+6π‘¦βˆ’3𝑧=π‘Ž. is {(0,𝑏,𝑐)}, find the values of π‘Ž, 𝑏, and 𝑐.

  • Aπ‘Ž=27, 𝑏=0, 𝑐=8
  • Bπ‘Ž=18, 𝑏=βˆ’4, 𝑐=3
  • Cπ‘Ž=0, 𝑏=6, 𝑐=βˆ’4
  • Dπ‘Ž=27, 𝑏=6, 𝑐=3
  • Eπ‘Ž=0, 𝑏=6, 𝑐=3

Q5:

Solve the simultaneous equations βˆ’3π‘₯βˆ’9π‘¦βˆ’2𝑧=118,βˆ’2π‘₯+6π‘¦βˆ’9𝑧=32,4π‘₯βˆ’8π‘¦βˆ’5𝑧=84.

  • Aπ‘₯=βˆ’824155, 𝑦=βˆ’1,234155, 𝑧=βˆ’33231
  • Bπ‘₯=0, 𝑦=βˆ’9, 𝑧=βˆ’50
  • Cπ‘₯=55017, 𝑦=βˆ’40934, 𝑧=βˆ’15417
  • Dπ‘₯=βˆ’7, 𝑦=βˆ’9, 𝑧=βˆ’8
  • Eπ‘₯=5, 𝑦=βˆ’2, 𝑧=βˆ’1352

Q6:

Solve the simultaneous equations 5𝑦+9𝑧=57,6π‘₯βˆ’7𝑧=3,5π‘₯+6𝑦=56.

  • Aπ‘₯=4, 𝑦=6, 𝑧=3
  • Bπ‘₯=3, 𝑦=0, 𝑧=157
  • Cπ‘₯=βˆ’272499, 𝑦=5,154499, 𝑧=297499
  • Dπ‘₯=0, 𝑦=βˆ’2, 𝑧=679
  • Eπ‘₯=3, 𝑦=27,959499, 𝑧=0

Q7:

Solve the simultaneous equations βˆ’4π‘₯+3π‘¦βˆ’6π‘§βˆ’12=0,7π‘₯+π‘¦βˆ’8π‘§βˆ’127=0,9π‘₯+8π‘¦βˆ’5π‘§βˆ’121=0.

  • Aπ‘₯=βˆ’2,721167, 𝑦=βˆ’9,324167, 𝑧=βˆ’8,770167
  • Bπ‘₯=βˆ’2, 𝑦=5, 𝑧=βˆ’5
  • Cπ‘₯=9, 𝑦=0, 𝑧=βˆ’8
  • Dπ‘₯=5, 𝑦=βˆ’6, 𝑧=112
  • Eπ‘₯=1,181149, 𝑦=4,192447, 𝑧=βˆ’1,408149

Q8:

Solve the simultaneous equations 9π‘₯+8𝑦+4𝑧=117,8π‘₯βˆ’2𝑦+7𝑧=70,2π‘₯βˆ’π‘¦=4.

  • Aπ‘₯=5, 𝑦=6, 𝑧=6
  • Bπ‘₯=7, 𝑦=6, 𝑧=0
  • Cπ‘₯=βˆ’5, 𝑦=3, 𝑧=31
  • Dπ‘₯=βˆ’34765, 𝑦=2,32265, 𝑧=12613
  • Eπ‘₯=1,355223, 𝑦=698223, 𝑧=3,338223

Q9:

Solve the simultaneous equations 2π‘₯+6𝑦+3𝑧=17,βˆ’9π‘₯βˆ’4𝑧=27,βˆ’3π‘₯+2𝑦=11.

  • Aπ‘₯=1, 𝑦=7, 𝑧=βˆ’9
  • Bπ‘₯=βˆ’14571, 𝑦=41671, 𝑧=βˆ’44171
  • Cπ‘₯=1, 𝑦=7, 𝑧=0
  • Dπ‘₯=βˆ’28155, 𝑦=1711, 𝑧=26155
  • Eπ‘₯=βˆ’3, 𝑦=0, 𝑧=βˆ’72

Q10:

Solve the simultaneous equations βˆ’9π‘₯βˆ’5π‘¦βˆ’π‘§=1,βˆ’4π‘₯βˆ’6π‘¦βˆ’5𝑧=1,4π‘₯βˆ’4𝑦+3𝑧=1.

  • Aπ‘₯=2231, 𝑦=131, 𝑧=531
  • Bπ‘₯=0, 𝑦=βˆ’419, 𝑧=119
  • Cπ‘₯=βˆ’29, 𝑦=βˆ’419, 𝑧=119
  • Dπ‘₯=βˆ’15, 𝑦=βˆ’419, 𝑧=119
  • Eπ‘₯=βˆ’113, 𝑦=βˆ’419, 𝑧=119

Q11:

Find the solution of the system of equations 65π‘₯+84𝑦+16𝑧=546, 81π‘₯+105𝑦+20𝑧=682, 84π‘₯+110𝑦+21𝑧=713, giving your answer in terms of an arbitrary real number 𝑑 if necessary.

  • Aπ‘₯=4, 𝑦=2, 𝑧=5
  • Bπ‘₯=2, 𝑦=4, 𝑧=5
  • Cπ‘₯=βˆ’2, 𝑦=4, 𝑧=5
  • Dπ‘₯=5, 𝑦=4, 𝑧=2
  • Eπ‘₯=2, 𝑦=6, 𝑧=βˆ’3

Q12:

Find the solution of the system of equations 9π‘₯βˆ’2𝑦+4𝑧=βˆ’17, 13π‘₯βˆ’3𝑦+6𝑧=βˆ’25, and βˆ’2π‘₯βˆ’π‘§=3, giving your answer in terms of an arbitrary real number 𝑑 if necessary.

  • Aπ‘₯=βˆ’1, 𝑦=βˆ’2, 𝑧=1
  • Bπ‘₯=1, 𝑦=βˆ’3, 𝑧=2
  • Cπ‘₯=1, 𝑦=βˆ’2, 𝑧=1
  • Dπ‘₯=βˆ’1, 𝑦=2, 𝑧=βˆ’1
  • Eπ‘₯=1, 𝑦=2, 𝑧=βˆ’1

Q13:

The sum of the length and width of a rectangular prism is 24 cm. Its width plus its height is 19 cm and the sum of its height and length is 31 cm. Calculate the volume of the rectangular prism.

Q14:

In the triangle 𝐴𝐡𝐢, one of the angles is the arithmetic mean of the other two. Find each angle of the triangle given the difference between the smaller and larger angles is 61∘.

  • A46∘, 64.5∘, 83∘
  • B98∘, 60∘, 22∘
  • C29.5∘, 120∘, 90.5∘
  • D90.5∘, 60∘, 29.5∘

Q15:

Solve the simultaneous equations 9π‘₯+8𝑦+6π‘§βˆ’5=0,7π‘₯βˆ’7π‘¦βˆ’4π‘§βˆ’44=0,9π‘₯βˆ’8π‘¦βˆ’π‘§βˆ’64=0.

  • Aπ‘₯=45529, 𝑦=βˆ’3,371145, 𝑧=6,483145
  • Bπ‘₯=βˆ’4, 𝑦=βˆ’8, 𝑧=βˆ’236
  • Cπ‘₯=3, 𝑦=βˆ’5, 𝑧=3
  • Dπ‘₯=βˆ’1, 𝑦=6, 𝑧=193
  • Eπ‘₯=βˆ’2,07777, 𝑦=βˆ’41, 𝑧=βˆ’1,09311

Q16:

Solve the simultaneous equations βˆ’7π‘₯+9π‘¦βˆ’8𝑧=19,4π‘₯βˆ’7𝑧=2,βˆ’3π‘₯+8𝑦=44.

  • Aπ‘₯=4, 𝑦=7, 𝑧=2
  • Bπ‘₯=βˆ’1,58053, 𝑦=βˆ’39753, 𝑧=1,03453
  • Cπ‘₯=4, 𝑦=7, 𝑧=0
  • Dπ‘₯=βˆ’41293, 𝑦=1,337279, 𝑧=βˆ’26293
  • Eπ‘₯=βˆ’1, 𝑦=0, 𝑧=βˆ’487

Q17:

Three numbers add up to 216. The sum of the first two numbers is 112 and the third number is 8 less than this sum. How many possible values are there for the numbers?

  • A0
  • B1
  • Cinfinitely many

Q18:

Find the values of π‘Ž, 𝑏, and 𝑐 given that (π‘₯+3), (π‘₯βˆ’2), and (π‘₯+4) are factors of π‘₯+π‘Žπ‘₯+𝑏π‘₯+π‘οŠ©οŠ¨.

  • Aπ‘Ž=5, 𝑏=βˆ’2, 𝑐=βˆ’24
  • Bπ‘Ž=5, 𝑏=2, 𝑐=βˆ’24
  • Cπ‘Ž=βˆ’5, 𝑏=2, 𝑐=24
  • Dπ‘Ž=βˆ’5, 𝑏=βˆ’2, 𝑐=βˆ’24
  • Eπ‘Ž=5, 𝑏=βˆ’2, 𝑐=24

Q19:

Four times the weight of Michael is 150 pounds more than the weight of Victoria. Four times the weight of Victoria is 660 pounds less than seventeen times the weight of Michael. Four times the weight of Michael plus the weight of Benjamin equals 290 pounds. Liam would balance all three of the others. Find the weights of the four people.

  • AMichael’s weight = 90 lb, Victoria’s weight = 60 lb, Liam’s weight = 200 lb, Benjamin’s weight = 50 lb
  • BMichael’s weight = 90 lb, Victoria’s weight = 60 lb, Liam’s weight = 50 lb, Benjamin’s weight = 200 lb
  • CMichael’s weight = 60 lb, Victoria’s weight = 90 lb, Liam’s weight = 200 lb, Benjamin’s weight = 50 lb
  • DMichael’s weight = 50 lb, Victoria’s weight = 60 lb, Liam’s weight = 90 lb, Benjamin’s weight = 200 lb
  • EMichael’s weight = 60 lb, Victoria’s weight = 90 lb, Liam’s weight = 50 lb, Benjamin’s weight = 200 lb

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