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.

  • A π‘₯ = 1 , 4 7 8 5 6 5 , 𝑦 = 8 7 7 5 6 5 , 𝑧 = βˆ’ 5 0 7 5 6 5
  • B π‘₯ = 7 5 2 3 1 5 , 𝑦 = 1 8 7 3 1 5 , 𝑧 = 5 1 8 9
  • C π‘₯ = 2 4 4 2 5 5 , 𝑦 = 1 8 1 8 5 , 𝑧 = βˆ’ 2 1 9 8 5
  • D π‘₯ = 6 5 , 𝑦 = 1 , 𝑧 = βˆ’ 3 5
  • E π‘₯ = 1 , 3 4 4 5 0 5 , 𝑦 = 2 2 1 1 0 1 , 𝑧 = βˆ’ 1 9 5 1 0 1

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.

  • A 34 years, 37 years, 43 years
  • B 34 years, 43 years, 46 years
  • C 37 years, 40 years, 46 years
  • D 37 years, 46 years, 49 years
  • E 34 years, 37 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.

  • A π‘Ž = βˆ’ 6 , 𝑏 = 7 , 𝑐 = 8
  • B π‘Ž = 1 4 6 8 7 , 𝑏 = 4 9 6 7 , 𝑐 = 2 0 9 5 3
  • C π‘Ž = βˆ’ 2 1 9 0 5 , 𝑏 = 4 6 4 7 , 𝑐 = βˆ’ 1 , 1 2 4 6 8 3
  • D π‘Ž = βˆ’ 3 5 , 𝑏 = βˆ’ 6 3 , 𝑐 = 1 9
  • E π‘Ž = 9 1 , 𝑏 = 6 3 , 𝑐 = 1 9

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.

  • A π‘Ž = 1 8 , 𝑏 = βˆ’ 4 , 𝑐 = 3
  • B π‘Ž = 0 , 𝑏 = 6 , 𝑐 = βˆ’ 4
  • C π‘Ž = 2 7 , 𝑏 = 0 , 𝑐 = 8
  • D π‘Ž = 2 7 , 𝑏 = 6 , 𝑐 = 3
  • E π‘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.

  • A π‘₯ = 0 , 𝑦 = βˆ’ 9 , 𝑧 = βˆ’ 5 0
  • B π‘₯ = βˆ’ 8 2 4 1 5 5 , 𝑦 = βˆ’ 1 , 2 3 4 1 5 5 , 𝑧 = βˆ’ 3 3 2 3 1
  • C π‘₯ = 5 5 0 1 7 , 𝑦 = βˆ’ 4 0 9 3 4 , 𝑧 = βˆ’ 1 5 4 1 7
  • D π‘₯ = 5 , 𝑦 = βˆ’ 2 , 𝑧 = βˆ’ 1 3 5 2
  • E π‘₯ = βˆ’ 7 , 𝑦 = βˆ’ 9 , 𝑧 = βˆ’ 8

Q6:

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

  • A π‘₯ = 4 , 𝑦 = 6 , 𝑧 = 3
  • B π‘₯ = 3 , 𝑦 = 0 , 𝑧 = 1 5 7
  • C π‘₯ = βˆ’ 2 7 2 4 9 9 , 𝑦 = 5 , 1 5 4 4 9 9 , 𝑧 = 2 9 7 4 9 9
  • D π‘₯ = 0 , 𝑦 = βˆ’ 2 , 𝑧 = 6 7 9
  • E π‘₯ = 3 , 𝑦 = 2 7 , 9 5 9 4 9 9 , 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.

  • A π‘₯ = 5 , 𝑦 = βˆ’ 6 , 𝑧 = 1 1 2
  • B π‘₯ = βˆ’ 2 , 𝑦 = 5 , 𝑧 = βˆ’ 5
  • C π‘₯ = 9 , 𝑦 = 0 , 𝑧 = βˆ’ 8
  • D π‘₯ = 1 , 1 8 1 1 4 9 , 𝑦 = 4 , 1 9 2 4 4 7 , 𝑧 = βˆ’ 1 , 4 0 8 1 4 9
  • E π‘₯ = βˆ’ 2 , 7 2 1 1 6 7 , 𝑦 = βˆ’ 9 , 3 2 4 1 6 7 , 𝑧 = βˆ’ 8 , 7 7 0 1 6 7

Q8:

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

  • A π‘₯ = βˆ’ 5 , 𝑦 = 3 , 𝑧 = 3 1
  • B π‘₯ = 5 , 𝑦 = 6 , 𝑧 = 6
  • C π‘₯ = 1 , 3 5 5 2 2 3 , 𝑦 = 6 9 8 2 2 3 , 𝑧 = 3 , 3 3 8 2 2 3
  • D π‘₯ = 7 , 𝑦 = 6 , 𝑧 = 0
  • E π‘₯ = βˆ’ 3 4 7 6 5 , 𝑦 = 2 , 3 2 2 6 5 , 𝑧 = 1 2 6 1 3

Q9:

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

  • A π‘₯ = 1 , 𝑦 = 7 , 𝑧 = βˆ’ 9
  • B π‘₯ = 1 , 𝑦 = 7 , 𝑧 = 0
  • C π‘₯ = βˆ’ 1 4 5 7 1 , 𝑦 = 4 1 6 7 1 , 𝑧 = βˆ’ 4 4 1 7 1
  • D π‘₯ = βˆ’ 3 , 𝑦 = 0 , 𝑧 = βˆ’ 7 2
  • E π‘₯ = βˆ’ 2 8 1 5 5 , 𝑦 = 1 7 1 1 , 𝑧 = 2 6 1 5 5

Q10:

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

  • A π‘₯ = 2 2 3 1 , 𝑦 = 1 3 1 , 𝑧 = 5 3 1
  • B π‘₯ = 0 , 𝑦 = βˆ’ 4 1 9 , 𝑧 = 1 1 9
  • C π‘₯ = βˆ’ 2 9 , 𝑦 = βˆ’ 4 1 9 , 𝑧 = 1 1 9
  • D π‘₯ = βˆ’ 1 5 , 𝑦 = βˆ’ 4 1 9 , 𝑧 = 1 1 9
  • E π‘₯ = βˆ’ 1 1 3 , 𝑦 = βˆ’ 4 1 9 , 𝑧 = 1 1 9

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.

  • A π‘₯ = 4 , 𝑦 = 2 , 𝑧 = 5
  • B π‘₯ = 2 , 𝑦 = 4 , 𝑧 = 5
  • C π‘₯ = βˆ’ 2 , 𝑦 = 4 , 𝑧 = 5
  • D π‘₯ = 5 , 𝑦 = 4 , 𝑧 = 2
  • E π‘₯ = 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.

  • A π‘₯ = βˆ’ 1 , 𝑦 = βˆ’ 2 , 𝑧 = 1
  • B π‘₯ = 1 , 𝑦 = βˆ’ 3 , 𝑧 = 2
  • C π‘₯ = 1 , 𝑦 = 2 , 𝑧 = βˆ’ 1
  • D π‘₯ = βˆ’ 1 , 𝑦 = 2 , 𝑧 = βˆ’ 1
  • E π‘₯ = 1 , 𝑦 = βˆ’ 2 , 𝑧 = 1

Q13:

The sum of the length and width of a cuboid 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 cuboid.

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∘.

  • A 2 9 . 5 ∘ , 1 2 0 ∘ , 9 0 . 5 ∘
  • B 4 6 ∘ , 6 4 . 5 ∘ , 8 3 ∘
  • C 9 0 . 5 ∘ , 6 0 ∘ , 2 9 . 5 ∘
  • D 9 8 ∘ , 6 0 ∘ , 2 2 ∘

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.

  • A π‘₯ = 4 5 5 2 9 , 𝑦 = βˆ’ 3 , 3 7 1 1 4 5 , 𝑧 = 6 , 4 8 3 1 4 5
  • B π‘₯ = βˆ’ 4 , 𝑦 = βˆ’ 8 , 𝑧 = βˆ’ 2 3 6
  • C π‘₯ = 3 , 𝑦 = βˆ’ 5 , 𝑧 = 3
  • D π‘₯ = βˆ’ 2 , 0 7 7 7 7 , 𝑦 = βˆ’ 4 1 , 𝑧 = βˆ’ 1 , 0 9 3 1 1
  • E π‘₯ = βˆ’ 1 , 𝑦 = 6 , 𝑧 = 1 9 3

Q16:

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

  • A π‘₯ = βˆ’ 4 1 2 9 3 , 𝑦 = 1 , 3 3 7 2 7 9 , 𝑧 = βˆ’ 2 6 2 9 3
  • B π‘₯ = βˆ’ 1 , 5 8 0 5 3 , 𝑦 = βˆ’ 3 9 7 5 3 , 𝑧 = 1 , 0 3 4 5 3
  • C π‘₯ = βˆ’ 1 , 𝑦 = 0 , 𝑧 = βˆ’ 4 8 7
  • D π‘₯ = 4 , 𝑦 = 7 , 𝑧 = 2
  • E π‘₯ = 4 , 𝑦 = 7 , 𝑧 = 0

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?

  • Ainfinitely many
  • B0
  • C1

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¨.

  • A π‘Ž = βˆ’ 5 , 𝑏 = βˆ’ 2 , 𝑐 = βˆ’ 2 4
  • B π‘Ž = 5 , 𝑏 = βˆ’ 2 , 𝑐 = 2 4
  • C π‘Ž = 5 , 𝑏 = βˆ’ 2 , 𝑐 = βˆ’ 2 4
  • D π‘Ž = 5 , 𝑏 = 2 , 𝑐 = βˆ’ 2 4
  • E π‘Ž = βˆ’ 5 , 𝑏 = 2 , 𝑐 = 2 4

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

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