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Lesson: Systems of Linear Equations in Three Variables

Sample Question Videos

Worksheet • 18 Questions • 1 Video

Q1:

Solve the simultaneous equations

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

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, 43 years, 46 years
  • B 37 years, 40 years, 46 years
  • C 34 years, 37 years, 43 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 is { ( βˆ’ 6 , 7 , 8 ) } , find the values of π‘Ž , 𝑏 , and 𝑐 .

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

Q4:

Given that the solution set of the simultaneous equations is { ( 0 , 𝑏 , 𝑐 ) } , find the values of π‘Ž , 𝑏 , and 𝑐 .

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

Q5:

Solve the simultaneous equations

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

Q6:

Solve the simultaneous equations

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

Q7:

Solve the simultaneous equations

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

Q8:

Solve the simultaneous equations

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

Q9:

Solve the simultaneous equations

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

Q10:

Solve the simultaneous equations

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

Q11:

Find the solution of the system of equations 6 5 π‘₯ + 8 4 𝑦 + 1 6 𝑧 = 5 4 6 , 8 1 π‘₯ + 1 0 5 𝑦 + 2 0 𝑧 = 6 8 2 , 8 4 π‘₯ + 1 1 0 𝑦 + 2 1 𝑧 = 7 1 3 , giving your answer in terms of an arbitrary real number 𝑑 if necessary.

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

Q12:

Find the solution of the system of equations 9 π‘₯ βˆ’ 2 𝑦 + 4 𝑧 = βˆ’ 1 7 , 1 3 π‘₯ βˆ’ 3 𝑦 + 6 𝑧 = βˆ’ 2 5 , and βˆ’ 2 π‘₯ βˆ’ 𝑧 = 3 , giving your answer in terms of an arbitrary real number 𝑑 if necessary.

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

Q13:

Find the solution of the system of equations 7 π‘₯ + 1 4 𝑦 + 1 5 𝑧 = 2 2 , 2 π‘₯ + 4 𝑦 + 3 𝑧 = 5 , 3 π‘₯ + 6 𝑦 + 1 0 𝑧 = 1 3 , giving your answer in terms of an arbitrary real number 𝑑 if necessary.

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

Q14:

Find the solution of the system of equations 3 π‘₯ βˆ’ 𝑦 + 4 𝑧 = 6 , 𝑦 + 8 𝑧 = 0 , βˆ’ 2 π‘₯ + 𝑦 = βˆ’ 4 , giving your answer in terms of an arbitrary real number 𝑑 if necessary.

  • A π‘₯ = 2 βˆ’ 4 𝑑 , 𝑧 = 𝑑 , 𝑦 = βˆ’ 8 𝑑
  • B π‘₯ = 2 + 3 𝑑 , 𝑧 = 2 𝑑 , 𝑦 = βˆ’ 8 𝑑
  • C π‘₯ = 2 + 4 𝑑 , 𝑧 = 𝑑 , 𝑦 = βˆ’ 8 𝑑
  • D π‘₯ = 2 + 4 𝑑 , 𝑧 = 𝑑 , 𝑦 = 8 𝑑
  • E π‘₯ = 2 βˆ’ 4 𝑑 , 𝑧 = 𝑑 , 𝑦 = 8 𝑑

Q15:

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.

Q16:

In the triangle 𝐴 𝐡 𝐢 , 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 6 1 ∘ .

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

Q17:

Solve the simultaneous equations

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

Q18:

Solve the simultaneous equations

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