Lesson Worksheet: Cramer’s Rule Mathematics

In this worksheet, we will practice using Cramer’s rule to solve a system of linear equations.

Q1:

Jacob is solving simultaneous equations using Cramer’s rule. He writes down Δ=||−9−82−2||,Δ=||9−9−32||.

What does he write down for Δ?

  • A||−9−82−2||
  • B||−992−3||
  • C||9−8−3−2||
  • D||−8−9−22||
  • E||9−9−32||

Q2:

Use determinants to solve the system 3𝑥+2𝑦−2𝑧−1=0,3𝑥+3𝑦−3𝑧+3=0,−2𝑥+4𝑦−5𝑧+1=0.

  • A𝑥=4, 𝑦=−4, 𝑧=32
  • B𝑥=35, 𝑦=−5, 𝑧=−65
  • C𝑥=−3, 𝑦=25, 𝑧=21
  • D𝑥=3, 𝑦=−25, 𝑧=−21
  • E𝑥=19, 𝑦=−21, 𝑧=−13

Q3:

Use determinants to solve the system 3𝑦=−3+5𝑧+5𝑥,5𝑥=4𝑦−3𝑧,2𝑥−2𝑦+2𝑧=2.

  • A𝑥=−2,𝑦=−1,𝑧=2
  • B𝑥=2,𝑦=1,𝑧=−2
  • C𝑥=12,𝑦=14,𝑧=−12
  • D𝑥=−3,𝑦=−4,𝑧=65
  • E𝑥=−2,𝑦=232,𝑧=−112

Q4:

Use determinants to solve the system −8𝑥−4𝑦=−8,9𝑥−6𝑦=−9.

  • A𝑥=127, 𝑦=17
  • B𝑥=17, 𝑦=127
  • C𝑥=1, 𝑦=12
  • D𝑥=−17, 𝑦=−127
  • E𝑥=6, 𝑦=−10

Q5:

Use determinants to solve the system −9𝑥=−8+8𝑦,6𝑦=7+3𝑥.

  • A𝑥=−439, 𝑦=2926
  • B𝑥=8, 𝑦=−8
  • C𝑥=2926, 𝑦=−439
  • D𝑥=439, 𝑦=−2926
  • E𝑥=−415, 𝑦=2910

Q6:

Use determinants to solve the system −7𝑥+2𝑦+5=0,𝑥+8𝑦−1=0.

  • A𝑥=2129, 𝑦=129
  • B𝑥=129, 𝑦=2129
  • C𝑥=1, 𝑦=1
  • D𝑥=79, 𝑦=127
  • E𝑥=−2129, 𝑦=−129

Q7:

Use determinants to solve the system 5𝑥=−2𝑦−5+3𝑧,−3𝑥−𝑦+1=2𝑧,2𝑦−𝑧=−5𝑥+3.

  • A𝑥=4, 𝑦=3, 𝑧=313
  • B𝑥=21, 𝑦=−56, 𝑧=−4
  • C𝑥=−21, 𝑦=56, 𝑧=4
  • D𝑥=2111, 𝑦=−5611, 𝑧=−811
  • E𝑥=−27, 𝑦=29, 𝑧=−26

Q8:

Solve, using Cramer's rule, the simultaneous equations |||−1𝑧−4𝑦|||=23,|||2𝑦−5𝑥|||=13,||3𝑥5𝑧||=51.

  • A𝑥=70253, 𝑦=−40353, 𝑧=7
  • B𝑥=66347, 𝑦=5, 𝑧=−37147
  • C𝑥=−6, 𝑦=5, 𝑧=7
  • D𝑥=−7, 𝑦=5, 𝑧=9
  • E𝑥=−6, 𝑦=7, 𝑧=6

Q9:

11 sketchbooks and 15 crayons cost 377 LE, and 15 sketchbooks and 16 crayons cost 474 LE. Use Cramer’s rule to find the price of a sketchbook and the price of a crayon.

  • AA sketchbook costs 9 LE and a crayon costs 2.70 LE.
  • BA sketchbook costs 22 LE and a crayon costs 9 LE.
  • CA sketchbook costs 441 LE and a crayon costs 9 LE.
  • DA sketchbook costs 22 LE and a crayon costs 1,078 LE.
  • EA sketchbook costs 441 LE and a crayon costs 1,078 LE.

Q10:

Use Cramer’s rule to solve the simultaneous equations |||−3𝑦2𝑥|||=−9,|||1𝑧1𝑦|||=−1,||4𝑥4𝑧||=2.

  • A𝑥=−2110, 𝑦=85, 𝑧=52
  • B𝑥=152, 𝑦=10, 𝑧=252
  • C𝑥=215, 𝑦=110, 𝑧=225
  • D𝑥=1, 𝑦=32, 𝑧=25
  • E𝑥=2, 𝑦=32, 𝑧=52

Q11:

Use determinants to solve the system 𝑥−5𝑦+3𝑧=5,3𝑥−4𝑦+2𝑧=−5,−𝑥+3𝑦−2𝑧=−5.

  • A𝑥=−53, 𝑦=203, 𝑧=403
  • B𝑥=53, 𝑦=−203, 𝑧=−403
  • C𝑥=−953, 𝑦=−703, 𝑧=1303
  • D𝑥=53, 𝑦=−203, 𝑧=10

Q12:

Use determinants to solve the system −2𝑥−2𝑦+4𝑧+2=0,5𝑥−4𝑦+2𝑧−5=0,2𝑥+3𝑦+𝑧−2=0.

  • A𝑥=9, 𝑦=−1, 𝑧=72
  • B𝑥=−5713, 𝑦=0, 𝑧=6913
  • C𝑥=−1, 𝑦=0, 𝑧=0
  • D𝑥=1, 𝑦=0, 𝑧=0
  • E𝑥=−119, 𝑦=−4057, 𝑧=1019

Q13:

Use determinants to solve the system 5𝑥+𝑦−2=0,9𝑥+4𝑦+1=0.

  • A𝑥=911, 𝑦=−2311
  • B𝑥=−2311, 𝑦=911
  • C𝑥=5, 𝑦=−23
  • D𝑥=929, 𝑦=−2329
  • E𝑥=−911, 𝑦=2311

Q14:

Use determinants to solve the system −5𝑥=−2−2𝑦,−6𝑦=−5+𝑥.

  • A𝑥=1116, 𝑦=2332
  • B𝑥=6, 𝑦=14
  • C𝑥=2332, 𝑦=1116
  • D𝑥=−1116, 𝑦=−2332
  • E𝑥=1114, 𝑦=2328

Q15:

True or False: Suppose that 𝑎𝑥+𝑏𝑦=𝑒,𝑐𝑥+𝑑𝑦=𝑓 is a system of equations with 𝑎𝑑−𝑏𝑐=1. Then 𝑦=𝑒𝑏𝑓𝑑.det

  • AFalse
  • BTrue

Q16:

Is Cramer’s rule useful for finding solutions to systems of linear equations in which there is an infinite set of solutions?

  • AYes
  • BNo

Q17:

True or False: In Cramer’s rule, 𝑥=ΔΔ.

  • AFalse
  • BTrue

Q18:

True or False: Cramer’s rule is used to determine solutions to linear systems when the determinant of the coefficient matrix equals zero.

  • ATrue
  • BFalse

Q19:

Noah is solving simultaneous equations using Cramer’s rule. He writes down the following: Δ=1234−3−221−4,Δ=213−34−2−12−4,Δ=221−3−34−112.

What does he write down for Δ?

  • A−11−4−3−34223
  • B223−3−3−2−11−4
  • C322−3−34−41−1
  • D−11−4−34−2223
  • E221−3−34−112

Q20:

Use Cramer’s rule to solve the linear system of equations that is in matrix form: 212−1𝑥𝑦=93.

  • A𝑥=3, 𝑦=3
  • B𝑥=−3, 𝑦=−3
  • C𝑥=3, 𝑦=−3
  • D𝑥=2, 𝑦=1
  • E𝑥=−3, 𝑦=3

Q21:

Complete the following: In Cramer’s rule, 𝑥=⋯Δ.

  • AΔ
  • BΔ
  • CΔ

Q22:

According to Cramer’s rule and given that Δ=||||5643−4−665−3||||,Δ=||||654−43−6−56−3||||,Δ=||||665−4−43−556||||,

write the simultaneous equations of the system.

  • A6𝑥+6𝑦+5𝑧=4, −4𝑥−4𝑦+3𝑧=−6, −5𝑥+5𝑦+6𝑧=−3
  • B6𝑥+6𝑦+4𝑧=0, −4𝑥−4𝑦−6𝑧=0, −5𝑥+5𝑦−3𝑧=0
  • C6𝑥+6𝑦+4𝑧=−5, −4𝑥−4𝑦−6𝑧=−3, −5𝑥+5𝑦−3𝑧=−6
  • D5𝑥+6𝑦+4𝑧=5, −6𝑥−4𝑦−6𝑧=3, −4𝑥−5𝑦−3𝑧=6
  • E6𝑥+6𝑦+4𝑧=5, −4𝑥−4𝑦−6𝑧=3, −5𝑥+5𝑦−3𝑧=6

Q23:

In Cramer’s rule, given that Δ=||902−5||,Δ=||−9982||, find the value of 𝑥 and 𝑦.

  • A𝑥=−1,
    𝑦=3
  • B𝑥=−1,
    𝑦=−2
  • C𝑥=1,
    𝑦=−2
  • D𝑥=−1,
    𝑦=2
  • E𝑥=2,
    𝑦=−1

Q24:

According to Cramer’s rule and given that Δ=||3752||,Δ=||−53−25||, write the simultaneous equations of the system.

  • A7𝑥−5𝑦=3,
    2𝑥+2𝑦=5
  • B−5𝑥+7𝑦=3,
    −2𝑥+2𝑦=5
  • C7𝑥−5𝑦=−3,
    2𝑥−2𝑦=−5
  • D−5𝑥+7𝑦=−3,
    −2𝑥+2𝑦=−5
  • E−5𝑥+7𝑦=0,
    −2𝑥+2𝑦=0

Q25:

Consider the following linear system of equations in matrix form: 1956−1−693−9𝑥𝑦𝑧=5−6−3.

Write the determinant Δ.

  • A||||595−6−1−6−33−9||||
  • B||||1556−6−69−3−9||||
  • C||||591−6−16−939||||
  • D||||1956−1−693−9||||
  • E||||1956−1−693−3||||

Write the determinant Δ.

  • A||||591−6−16−939||||
  • B||||1956−1−693−3||||
  • C||||1556−6−69−3−9||||
  • D||||595−6−1−6−33−9||||
  • E||||1956−1−693−9||||

Write the determinant Δ.

  • A||||1956−1−693−3||||
  • B||||1956−1−693−9||||
  • C||||595−6−1−6−33−9||||
  • D||||1556−6−69−3−9||||
  • E||||591−6−16−939||||

Write the determinant Δ.

  • A||||1956−1−693−3||||
  • B||||595−6−1−6−33−9||||
  • C||||591−6−16−939||||
  • D||||1956−1−693−9||||
  • E||||1556−6−69−3−9||||

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