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

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.

  • A𝑥=11, 𝑦=9, 𝑧=17
  • B𝑥=9, 𝑦=11, 𝑧=17
  • C𝑥=10, 𝑦=16, 𝑧=17
  • D𝑥=16, 𝑦=10, 𝑧=17
  • E𝑥=17, 𝑦=10, 𝑧=16

Q2:

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

  • A𝑥=16, 𝑦=8, 𝑧=4
  • B𝑥=8, 𝑦=4, 𝑧=16
  • C𝑥=4, 𝑦=8, 𝑧=16
  • D𝑥=16, 𝑦=4, 𝑧=8

Q3:

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

  • A𝑥=3, 𝑦=4, 𝑧=5
  • B𝑥=4, 𝑦=5, 𝑧=3
  • C𝑥=5, 𝑦=4, 𝑧=3
  • D𝑥=3, 𝑦=5, 𝑧=4

Q4:

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

  • A𝑥=2, 𝑦=1, 𝑧=5
  • B𝑥=1, 𝑦=5, 𝑧=2
  • C𝑥=5, 𝑦=1, 𝑧=2
  • D𝑥=2, 𝑦=5, 𝑧=1

Q5:

Given that 113025301=2111585632,

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

  • A𝑋=210291510683622
  • B𝑋=107373482228199
  • C𝑋=556322841131116
  • D𝑋=169595126148
  • E𝑋=556322841131116

Q6:

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

  • A𝑥=1, 𝑦=0, 𝑧=7
  • B𝑥=0, 𝑦=1, 𝑧=7
  • C𝑥=7, 𝑦=0, 𝑧=1
  • D𝑥=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.

  • A𝑥=24, 𝑦=41, 𝑧=12
  • B𝑥=12, 𝑦=24, 𝑧=41
  • C𝑥=41, 𝑦=24, 𝑧=12
  • D𝑥=12, 𝑦=41, 𝑧=24

Q8:

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

  • A𝑥𝑦𝑧=5268138
  • B𝑥𝑦𝑧=11732641,770878
  • C𝑥𝑦𝑧=11731,2661,7961,120
  • D𝑥𝑦𝑧=1173792196210
  • E𝑥𝑦𝑧=166332838

Q9:

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

Express the system as a single matrix equation.

  • A124110348𝑝𝑞𝑟=61110
  • B124101348𝑝𝑞𝑟=11610
  • C113204418𝑝𝑞𝑟=11610
  • D113204418𝑝𝑞𝑟=61110
  • E124110348𝑝𝑞𝑟=11610

Work out the inverse of the coefficient matrix.

  • A12883228828281103
  • B142432211454102
  • C142411432410252
  • D142411432410252
  • E12888132281028283

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

  • A𝑝𝑞𝑟=12802419
  • B𝑝𝑞𝑟=142256954
  • C𝑝𝑞𝑟=1426242832
  • D𝑝𝑞𝑟=1281207674
  • E𝑝𝑞𝑟=14213724847

Q10:

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

Express the system as a single matrix equation.

  • A142215312𝑎𝑏𝑐=5711
  • B142215312𝑎𝑏𝑐=5711
  • C142215312𝑎𝑏𝑐=7511
  • D123411252𝑎𝑏𝑐=5711
  • E123411252𝑎𝑏𝑐=7511

Work out the inverse of the coefficient matrix.

  • A1493111104132219
  • B1433102211891117
  • C1433111108112297
  • D1433111108112297
  • E1493102211411139

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

  • A𝑎𝑏𝑐=14365231276
  • B𝑎𝑏𝑐=143340220140
  • C𝑎𝑏𝑐=14965233248
  • D𝑎𝑏𝑐=14981221202
  • E𝑎𝑏𝑐=143327210149

Q11:

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

Express the system as a single matrix equation.

  • A212215416𝑝𝑞𝑟=41410
  • B224111256𝑝𝑞𝑟=41410
  • C212215416𝑝𝑞𝑟=41410
  • D224111256𝑝𝑞𝑟=14410
  • E212215416𝑝𝑞𝑟=14410

Work out the inverse of the coefficient matrix.

  • A16143846220
  • B16143846220
  • C16182442360
  • D110182442360
  • E110187846264

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

  • A𝑝𝑞𝑟=13111410
  • B𝑝𝑞𝑟=1523626
  • C𝑝𝑞𝑟=1512146
  • D𝑝𝑞𝑟=13143410
  • E𝑝𝑞𝑟=13642648

Q12:

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

  • A𝑥𝑦𝑧=12621156
  • B𝑥𝑦𝑧=10013151
  • C𝑥𝑦𝑧=126386939
  • D𝑥𝑦𝑧=152218365143
  • E𝑥𝑦𝑧=1782543459

Q13:

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

  • A1621557
  • B72220
  • C72220
  • D72230
  • E72220

Q14:

True or False: If the equations of three nonintersecting planes are represented in the matrix form 𝐴𝑋=𝐵, then matrix 𝐴 can be inverted.

  • AFalse
  • BTrue

Q15:

True or False: The solution of the matrix equation 𝐴𝑋=𝐵 is 𝑋=𝐴𝐵.

  • ATrue
  • BFalse

Q16:

Consider the following matrix equation: 7578107977𝑥𝑦𝑧=12𝑘.

Find the value of 𝑘 that results in 𝑥=12.

Q17:

Consider the following matrix equation: 631406221𝑥𝑦𝑧=3𝑘5.

Find the value of 𝑘 that results in 𝑦=3116.

Q18:

True or False: In the matrix equation 𝐴𝑋=𝐵, the existence of a solution depends only on the 𝐴 matrix.

  • AFalse
  • BTrue

Q19:

True or False: In the matrix equation 𝐴𝑋=𝐵the value of the solution, if it exists, depends only on matrix 𝐴.

  • ATrue
  • BFalse

Q20:

True or False: If the coefficient matrix is invertible, then the system of equations has a unique solution.

  • ATrue
  • BFalse

Q21:

Consider the matrix equation 122312112𝑥𝑦𝑧=𝑚𝑘0.

Find the solution in terms of the constants 𝑘 and 𝑚.

  • A𝑥=𝑚3,𝑦=𝑘2+𝑚3,𝑧=𝑘4+𝑚3
  • B𝑥=𝑘2+𝑚3,𝑦=𝑘4+𝑚3,𝑧=𝑚3
  • C𝑥=𝑘2+𝑚3,𝑦=𝑚3,𝑧=𝑘4+𝑚3
  • D𝑥=𝑘4+𝑚3,𝑦=𝑘2+𝑚3,𝑧=𝑚3
  • E𝑥=𝑘2+𝑚3,𝑦=𝑘4+𝑚3,𝑧=𝑘2+𝑚3

Q22:

Consider the matrix equation 432311321𝑥𝑦𝑧=2𝑘3.

Find the solution in terms of the constant 𝑘.

  • A𝑥=7𝑘+1332, 𝑦=5𝑘916, 𝑧=21𝑘32
  • B𝑥=7𝑘+1332, 𝑦=21𝑘32, 𝑧=5𝑘916
  • C𝑥=7𝑘+1332, 𝑦=21𝑘32, 𝑧=7𝑘+1332
  • D𝑥=21𝑘32, 𝑦=7𝑘+1332, 𝑧=5𝑘916
  • E𝑥=5𝑘916, 𝑦=7𝑘+1332, 𝑧=21𝑘32

Q23:

𝛼, 𝛽, and 𝛾 are angles of a triangle. 2𝛽+𝛼=72 and 𝛼+12𝛾=90+32𝛽. Find the three angles.

  • A𝛼=120, 𝛽=12, and 𝛾=48
  • B𝛼=48, 𝛽=120, and 𝛾=12
  • C𝛼=12, 𝛽=48, and 𝛾=120
  • D𝛼=48, 𝛽=12, and 𝛾=120
  • E𝛼=12, 𝛽=120, and 𝛾=48

Q24:

Consider the matrix equation 0022𝑚1312𝑥𝑦𝑧=100.

Find the solution in terms of the constant 𝑚.

  • A𝑥=72(3𝑚2), 𝑦=2𝑚+16𝑚4, 𝑧=12
  • B𝑥=12, 𝑦=2𝑚+16𝑚4, 𝑧=72(3𝑚2)
  • C𝑥=2𝑚+16𝑚4, 𝑦=72(3𝑚2), 𝑧=12
  • D𝑥=2𝑚+16𝑚4, 𝑦=12, 𝑧=72(3𝑚2)
  • E𝑥=72(3𝑚2), 𝑦=12, 𝑧=2𝑚+16𝑚4

Q25:

Three planes are defined by the following equations: 3𝑥𝑦2𝑧2=0, 4𝑥𝑦3𝑧+1=0, and 4𝑥+3𝑦+𝑧4=0. Find their point of intersection.

  • A𝑥=2548, 𝑦=4148, 𝑧=3148
  • B𝑥=4148, 𝑦=2548, 𝑧=3148
  • C𝑥=2548, 𝑦=4148, 𝑧=3148
  • D𝑥=4148, 𝑦=3148, 𝑧=2548
  • E𝑥=2548, 𝑦=3148, 𝑧=4148

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