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=||9822||,Δ=||9932||.

What does he write down for Δ?

  • A||9822||
  • B||9923||
  • C||9832||
  • D||8922||
  • E||9932||

Q2:

Use determinants to solve the system 3𝑥+2𝑦2𝑧1=0,3𝑥+3𝑦3𝑧+3=0,2𝑥+4𝑦5𝑧+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=4𝑦3𝑧,2𝑥2𝑦+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 8𝑥4𝑦=8,9𝑥6𝑦=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+8𝑦1=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=2𝑦5+3𝑧,3𝑥𝑦+1=2𝑧,2𝑦𝑧=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 |||1𝑧4𝑦|||=23,|||2𝑦5𝑥|||=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 𝑥5𝑦+3𝑧=5,3𝑥4𝑦+2𝑧=5,𝑥+3𝑦2𝑧=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 2𝑥2𝑦+4𝑧+2=0,5𝑥4𝑦+2𝑧5=0,2𝑥+3𝑦+𝑧2=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+𝑦2=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=22𝑦,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 𝑎𝑑𝑏𝑐=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=123432214,Δ=213342124,Δ=221334112.

What does he write down for Δ?

  • A114334223
  • B223332114
  • C322334411
  • D114342223
  • E221334112

Q20:

Use Cramer’s rule to solve the linear system of equations that is in matrix form: 2121𝑥𝑦=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=||||564346653||||,Δ=||||654436563||||,Δ=||||665443556||||,

write the simultaneous equations of the system.

  • A6𝑥+6𝑦+5𝑧=4, 4𝑥4𝑦+3𝑧=6, 5𝑥+5𝑦+6𝑧=3
  • B6𝑥+6𝑦+4𝑧=0, 4𝑥4𝑦6𝑧=0, 5𝑥+5𝑦3𝑧=0
  • C6𝑥+6𝑦+4𝑧=5, 4𝑥4𝑦6𝑧=3, 5𝑥+5𝑦3𝑧=6
  • D5𝑥+6𝑦+4𝑧=5, 6𝑥4𝑦6𝑧=3, 4𝑥5𝑦3𝑧=6
  • E6𝑥+6𝑦+4𝑧=5, 4𝑥4𝑦6𝑧=3, 5𝑥+5𝑦3𝑧=6

Q23:

In Cramer’s rule, given that Δ=||9025||,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=||5325||, write the simultaneous equations of the system.

  • A7𝑥5𝑦=3,
    2𝑥+2𝑦=5
  • B5𝑥+7𝑦=3,
    2𝑥+2𝑦=5
  • C7𝑥5𝑦=3,
    2𝑥2𝑦=5
  • D5𝑥+7𝑦=3,
    2𝑥+2𝑦=5
  • E5𝑥+7𝑦=0,
    2𝑥+2𝑦=0

Q25:

Consider the following linear system of equations in matrix form: 195616939𝑥𝑦𝑧=563.

Write the determinant Δ.

  • A||||595616339||||
  • B||||155666939||||
  • C||||591616939||||
  • D||||195616939||||
  • E||||195616933||||

Write the determinant Δ.

  • A||||591616939||||
  • B||||195616933||||
  • C||||155666939||||
  • D||||595616339||||
  • E||||195616939||||

Write the determinant Δ.

  • A||||195616933||||
  • B||||195616939||||
  • C||||595616339||||
  • D||||155666939||||
  • E||||591616939||||

Write the determinant Δ.

  • A||||195616933||||
  • B||||595616339||||
  • C||||591616939||||
  • D||||195616939||||
  • E||||155666939||||

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