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

  • A 𝑥 = 7 5 2 3 1 5 , 𝑦 = 1 8 7 3 1 5 , 𝑧 = 5 1 8 9
  • B 𝑥 = 2 4 4 2 5 5 , 𝑦 = 1 8 1 8 5 , 𝑧 = 2 1 9 8 5
  • C 𝑥 = 1 3 4 4 5 0 5 , 𝑦 = 2 2 1 1 0 1 , 𝑧 = 1 9 5 1 0 1
  • D 𝑥 = 6 5 , 𝑦 = 1 , 𝑧 = 3 5
  • E 𝑥 = 1 4 7 8 5 6 5 , 𝑦 = 8 7 7 5 6 5 , 𝑧 = 5 0 7 5 6 5

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

Q3:

Given that the solution set of the simultaneous equations is { ( 6 , 7 , 8 ) } , find the values of 𝑎 , 𝑏 , and 𝑐 .

  • A 𝑎 = 3 5 , 𝑏 = 6 3 , 𝑐 = 1 9
  • B 𝑎 = 6 , 𝑏 = 7 , 𝑐 = 8
  • C 𝑎 = 2 1 9 0 5 , 𝑏 = 4 6 4 7 , 𝑐 = 1 1 2 4 6 8 3
  • D 𝑎 = 9 1 , 𝑏 = 6 3 , 𝑐 = 1 9
  • E 𝑎 = 1 4 6 8 7 , 𝑏 = 4 9 6 7 , 𝑐 = 2 0 9 5 3

Q4:

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

  • A 𝑎 = 0 , 𝑏 = 6 , 𝑐 = 4
  • B 𝑎 = 1 8 , 𝑏 = 4 , 𝑐 = 3
  • C 𝑎 = 0 , 𝑏 = 6 , 𝑐 = 3
  • D 𝑎 = 2 7 , 𝑏 = 6 , 𝑐 = 3
  • E 𝑎 = 2 7 , 𝑏 = 0 , 𝑐 = 8

Q5:

Solve the simultaneous equations

  • A 𝑥 = 0 , 𝑦 = 9 , 𝑧 = 5 0
  • B 𝑥 = 5 , 𝑦 = 2 , 𝑧 = 1 3 5 2
  • C 𝑥 = 8 2 4 1 5 5 , 𝑦 = 1 2 3 4 1 5 5 , 𝑧 = 3 3 2 3 1
  • D 𝑥 = 7 , 𝑦 = 9 , 𝑧 = 8
  • E 𝑥 = 5 5 0 1 7 , 𝑦 = 4 0 9 3 4 , 𝑧 = 1 5 4 1 7

Q6:

Solve the simultaneous equations

  • A 𝑥 = 3 , 𝑦 = 0 , 𝑧 = 1 5 7
  • B 𝑥 = 0 , 𝑦 = 2 , 𝑧 = 6 7 9
  • C 𝑥 = 3 , 𝑦 = 2 7 9 5 9 4 9 9 , 𝑧 = 0
  • D 𝑥 = 4 , 𝑦 = 6 , 𝑧 = 3
  • E 𝑥 = 2 7 2 4 9 9 , 𝑦 = 5 1 5 4 4 9 9 , 𝑧 = 2 9 7 4 9 9

Q7:

Solve the simultaneous equations

  • A 𝑥 = 5 , 𝑦 = 6 , 𝑧 = 1 1 2
  • B 𝑥 = 2 , 𝑦 = 5 , 𝑧 = 5
  • C 𝑥 = 2 7 2 1 1 6 7 , 𝑦 = 9 3 2 4 1 6 7 , 𝑧 = 8 7 7 0 1 6 7
  • D 𝑥 = 9 , 𝑦 = 0 , 𝑧 = 8
  • E 𝑥 = 1 1 8 1 1 4 9 , 𝑦 = 4 1 9 2 4 4 7 , 𝑧 = 1 4 0 8 1 4 9

Q8:

Solve the simultaneous equations

  • A 𝑥 = 5 , 𝑦 = 3 , 𝑧 = 3 1
  • B 𝑥 = 7 , 𝑦 = 6 , 𝑧 = 0
  • C 𝑥 = 1 3 5 5 2 2 3 , 𝑦 = 6 9 8 2 2 3 , 𝑧 = 3 3 3 8 2 2 3
  • D 𝑥 = 5 , 𝑦 = 6 , 𝑧 = 6
  • E 𝑥 = 3 4 7 6 5 , 𝑦 = 2 3 2 2 6 5 , 𝑧 = 1 2 6 1 3

Q9:

Solve the simultaneous equations

  • A 𝑥 = 3 , 𝑦 = 0 , 𝑧 = 7 2
  • B 𝑥 = 1 , 𝑦 = 7 , 𝑧 = 0
  • C 𝑥 = 1 4 5 7 1 , 𝑦 = 4 1 6 7 1 , 𝑧 = 4 4 1 7 1
  • D 𝑥 = 1 , 𝑦 = 7 , 𝑧 = 9
  • E 𝑥 = 2 8 1 5 5 , 𝑦 = 1 7 1 1 , 𝑧 = 2 6 1 5 5

Q10:

Solve the simultaneous equations

  • A 𝑥 = 2 2 3 1 , 𝑦 = 1 3 1 , 𝑧 = 5 3 1
  • B 𝑥 = 1 1 3 , 𝑦 = 4 1 9 , 𝑧 = 1 1 9
  • C 𝑥 = 2 9 , 𝑦 = 4 1 9 , 𝑧 = 1 1 9
  • D 𝑥 = 0 , 𝑦 = 4 1 9 , 𝑧 = 1 1 9
  • E 𝑥 = 1 5 , 𝑦 = 4 1 9 , 𝑧 = 1 1 9

Q11:

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

  • A 𝑥 = 5 , 𝑦 = 4 , 𝑧 = 2
  • B 𝑥 = 4 , 𝑦 = 2 , 𝑧 = 5
  • C 𝑥 = 2 , 𝑦 = 4 , 𝑧 = 5
  • D 𝑥 = 2 , 𝑦 = 4 , 𝑧 = 5
  • E 𝑥 = 2 , 𝑦 = 6 , 𝑧 = 3

Q12:

Find the solution of the system of equations 9 𝑥 2 𝑦 + 4 𝑧 = 1 7 , 1 3 𝑥 3 𝑦 + 6 𝑧 = 2 5 , and 2 𝑥 𝑧 = 3 , giving your answer in terms of an arbitrary real number 𝑡 if necessary.

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

Q13:

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

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

Q14:

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

  • A 𝑥 = 2 4 𝑡 , 𝑧 = 𝑡 , 𝑦 = 8 𝑡
  • B 𝑥 = 2 + 4 𝑡 , 𝑧 = 𝑡 , 𝑦 = 8 𝑡
  • C 𝑥 = 2 + 4 𝑡 , 𝑧 = 𝑡 , 𝑦 = 8 𝑡
  • D 𝑥 = 2 4 𝑡 , 𝑧 = 𝑡 , 𝑦 = 8 𝑡
  • E 𝑥 = 2 + 3 𝑡 , 𝑧 = 2 𝑡 , 𝑦 = 8 𝑡

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

Q17:

Solve the simultaneous equations

  • A 𝑥 = 1 , 𝑦 = 6 , 𝑧 = 1 9 3
  • B 𝑥 = 4 , 𝑦 = 8 , 𝑧 = 2 3 6
  • C 𝑥 = 4 5 5 2 9 , 𝑦 = 3 3 7 1 1 4 5 , 𝑧 = 6 4 8 3 1 4 5
  • D 𝑥 = 3 , 𝑦 = 5 , 𝑧 = 3
  • E 𝑥 = 2 0 7 7 7 7 , 𝑦 = 4 1 , 𝑧 = 1 0 9 3 1 1

Q18:

Solve the simultaneous equations

  • A 𝑥 = 1 , 𝑦 = 0 , 𝑧 = 4 8 7
  • B 𝑥 = 4 , 𝑦 = 7 , 𝑧 = 0
  • C 𝑥 = 1 5 8 0 5 3 , 𝑦 = 3 9 7 5 3 , 𝑧 = 1 0 3 4 5 3
  • D 𝑥 = 4 , 𝑦 = 7 , 𝑧 = 2
  • E 𝑥 = 4 1 2 9 3 , 𝑦 = 1 3 3 7 2 7 9 , 𝑧 = 2 6 2 9 3

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