Worksheet: Solving a System of Three Equations Using a Matrix Inverse

In this worksheet, we will practice solving a system of three linear equations using the inverse of the matrix of coefficients.

Q1:

Solve 111111110𝑥𝑦𝑧=9116 using the inverse of a matrix.

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

Q2:

Given that 111200840𝑥𝑦𝑧=480, find the values of 𝑥, 𝑦, and 𝑧.

  • A 𝑥 = 1 6 , 𝑦 = 8 , 𝑧 = 4
  • B 𝑥 = 8 , 𝑦 = 4 , 𝑧 = 1 6
  • C 𝑥 = 4 , 𝑦 = 8 , 𝑧 = 1 6
  • D 𝑥 = 1 6 , 𝑦 = 4 , 𝑧 = 8

Q3:

Given that 166024247𝑥𝑦𝑧=829, find the values of 𝑥, 𝑦, and 𝑧.

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

Q4:

Given that 930924525𝑥𝑦𝑧=695, find the values of 𝑥, 𝑦, and 𝑧.

  • A 𝑥 = 2 , 𝑦 = 1 , 𝑧 = 5
  • B 𝑥 = 1 , 𝑦 = 5 , 𝑧 = 2
  • C 𝑥 = 5 , 𝑦 = 1 , 𝑧 = 2
  • D 𝑥 = 2 , 𝑦 = 5 , 𝑧 = 1

Q5:

Given that 113025301=2111585632,

solve the following matrix equation for 𝑋: 123701022113025301𝑋=122611220.

  • A 𝑋 = 2 1 0 2 9 1 5 1 0 6 8 3 6 2 2
  • B 𝑋 = 1 0 7 3 7 3 4 8 2 2 2 8 1 9 9
  • C 𝑋 = 5 5 6 3 2 2 8 4 1 1 3 1 1 1 6
  • D 𝑋 = 1 6 9 5 9 5 1 2 6 1 4 8
  • E 𝑋 = 5 5 6 3 2 2 8 4 1 1 3 1 1 1 6

Q6:

Solve the system of the linear equations 𝑥+𝑦+𝑧=8, 2𝑥+𝑦𝑧=5, and 6𝑥3𝑦=6 using the inverse of a matrix.

  • A 𝑥 = 1 , 𝑦 = 0 , 𝑧 = 7
  • B 𝑥 = 0 , 𝑦 = 1 , 𝑧 = 7
  • C 𝑥 = 7 , 𝑦 = 0 , 𝑧 = 1
  • D 𝑥 = 7 , 𝑦 = 1 , 𝑧 = 0

Q7:

Use the inverse of a matrix to solve the system of linear equations 4𝑥2𝑦9𝑧=8, 3𝑥2𝑦6𝑧=3, and 𝑥+𝑦6𝑧=7.

  • A 𝑥 = 2 4 , 𝑦 = 4 1 , 𝑧 = 1 2
  • B 𝑥 = 1 2 , 𝑦 = 2 4 , 𝑧 = 4 1
  • C 𝑥 = 4 1 , 𝑦 = 2 4 , 𝑧 = 1 2
  • D 𝑥 = 1 2 , 𝑦 = 4 1 , 𝑧 = 2 4

Q8:

Use matrices to solve the following system of equations: 𝑥+8𝑦3𝑧=10,4𝑥3𝑦+8𝑧=12,6𝑥12𝑦+19𝑧=18.

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

Q9:

Consider the system of equations 𝑝2𝑞4𝑟=11𝑝+𝑟=63𝑝+4𝑞8𝑟=10.

Express the system as a single matrix equation.

  • A 1 2 4 1 1 0 3 4 8 𝑝 𝑞 𝑟 = 6 1 1 1 0
  • B 1 2 4 1 0 1 3 4 8 𝑝 𝑞 𝑟 = 1 1 6 1 0
  • C 1 1 3 2 0 4 4 1 8 𝑝 𝑞 𝑟 = 1 1 6 1 0
  • D 1 1 3 2 0 4 4 1 8 𝑝 𝑞 𝑟 = 6 1 1 1 0
  • E 1 2 4 1 1 0 3 4 8 𝑝 𝑞 𝑟 = 1 1 6 1 0

Work out the inverse of the coefficient matrix.

  • A 1 2 8 8 3 2 2 8 8 2 8 2 8 1 1 0 3
  • B 1 4 2 4 3 2 2 1 1 4 5 4 1 0 2
  • C 1 4 2 4 1 1 4 3 2 4 1 0 2 5 2
  • D 1 4 2 4 1 1 4 3 2 4 1 0 2 5 2
  • E 1 2 8 8 8 1 3 2 2 8 1 0 2 8 2 8 3

Multiply through by the inverse, on the left-hand side, to solve the matrix equation.

  • A 𝑝 𝑞 𝑟 = 1 2 8 0 2 4 1 9
  • B 𝑝 𝑞 𝑟 = 1 4 2 2 5 6 9 5 4
  • C 𝑝 𝑞 𝑟 = 1 4 2 6 2 4 2 8 3 2
  • D 𝑝 𝑞 𝑟 = 1 2 8 1 2 0 7 6 7 4
  • E 𝑝 𝑞 𝑟 = 1 4 2 1 3 7 2 4 8 4 7

Q10:

Consider the system of equations 2𝑐=5+𝑎4𝑏2𝑎+𝑏=75𝑐2𝑐=11+𝑏3𝑎.

Express the system as a single matrix equation.

  • A 1 4 2 2 1 5 3 1 2 𝑎 𝑏 𝑐 = 5 7 1 1
  • B 1 4 2 2 1 5 3 1 2 𝑎 𝑏 𝑐 = 5 7 1 1
  • C 1 4 2 2 1 5 3 1 2 𝑎 𝑏 𝑐 = 7 5 1 1
  • D 1 2 3 4 1 1 2 5 2 𝑎 𝑏 𝑐 = 5 7 1 1
  • E 1 2 3 4 1 1 2 5 2 𝑎 𝑏 𝑐 = 7 5 1 1

Work out the inverse of the coefficient matrix.

  • A 1 4 9 3 1 1 1 1 0 4 1 3 2 2 1 9
  • B 1 4 3 3 1 0 2 2 1 1 8 9 1 1 1 7
  • C 1 4 3 3 1 1 1 1 0 8 1 1 2 2 9 7
  • D 1 4 3 3 1 1 1 1 0 8 1 1 2 2 9 7
  • E 1 4 9 3 1 0 2 2 1 1 4 1 1 1 3 9

Multiply through by the inverse, on the left-hand side, to solve the matrix equation.

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

Q11:

Consider the system of equations 2𝑝+2𝑞+4𝑟=4𝑝𝑞𝑟=142𝑝+5𝑞+6𝑟=10.

Express the system as a single matrix equation.

  • A 2 1 2 2 1 5 4 1 6 𝑝 𝑞 𝑟 = 4 1 4 1 0
  • B 2 2 4 1 1 1 2 5 6 𝑝 𝑞 𝑟 = 4 1 4 1 0
  • C 2 1 2 2 1 5 4 1 6 𝑝 𝑞 𝑟 = 4 1 4 1 0
  • D 2 2 4 1 1 1 2 5 6 𝑝 𝑞 𝑟 = 1 4 4 1 0
  • E 2 1 2 2 1 5 4 1 6 𝑝 𝑞 𝑟 = 1 4 4 1 0

Work out the inverse of the coefficient matrix.

  • A 1 6 1 4 3 8 4 6 2 2 0
  • B 1 6 1 4 3 8 4 6 2 2 0
  • C 1 6 1 8 2 4 4 2 3 6 0
  • D 1 1 0 1 8 2 4 4 2 3 6 0
  • E 1 1 0 1 8 7 8 4 6 2 6 4

Multiply through by the inverse, on the left-hand side, to solve the matrix equation.

  • A 𝑝 𝑞 𝑟 = 1 3 1 1 1 4 1 0
  • B 𝑝 𝑞 𝑟 = 1 5 2 3 6 2 6
  • C 𝑝 𝑞 𝑟 = 1 5 1 2 1 4 6
  • D 𝑝 𝑞 𝑟 = 1 3 1 4 3 4 1 0
  • E 𝑝 𝑞 𝑟 = 1 3 6 4 2 6 4 8

Q12:

Use matrices to solve the following system of equations: 4𝑥+𝑦3𝑧=13,3𝑥+4𝑦2𝑧=18,5𝑥2𝑦+8𝑧=10.

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

Q13:

Use the inverse matrix to solve 234556789𝑥𝑦𝑧=04545, giving your answer as an appropriate matrix.

  • A 1 6 2 1 5 5 7
  • B 7 2 2 2 0
  • C 7 2 2 2 0
  • D 7 2 2 3 0
  • E 7 2 2 2 0

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.