Worksheet: Cramer's Rule

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

Q1:

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

  • Ayes
  • Bno

Q2:

William 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 − 8 2 − 2 | |
  • B | | − 9 9 2 − 3 | |
  • C | | 9 − 8 − 3 − 2 | |
  • D | | − 8 − 9 − 2 2 | |
  • E | | 9 − 9 − 3 2 | |

Q3:

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

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

Q4:

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

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

Q5:

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

  • A 𝑥 = 1 2 7 , 𝑦 = 1 7
  • B 𝑥 = 1 7 , 𝑦 = 1 2 7
  • C 𝑥 = 1 , 𝑦 = 1 2
  • D 𝑥 = − 1 7 , 𝑦 = − 1 2 7
  • E 𝑥 = 6 , 𝑦 = − 1 0

Q6:

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

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

Q7:

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

  • A 𝑥 = 2 1 2 9 , 𝑦 = 1 2 9
  • B 𝑥 = 1 2 9 , 𝑦 = 2 1 2 9
  • C 𝑥 = 1 , 𝑦 = 1
  • D 𝑥 = 7 9 , 𝑦 = 1 2 7
  • E 𝑥 = − 2 1 2 9 , 𝑦 = − 1 2 9

Q8:

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

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

Q9:

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

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

Q10:

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.

Q11:

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

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

Q12:

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

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

Q13:

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

  • AFalse
  • BTrue

Q14:

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

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

Q15:

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

  • A 𝑥 = 9 1 1 , 𝑦 = − 2 3 1 1
  • B 𝑥 = − 2 3 1 1 , 𝑦 = 9 1 1
  • C 𝑥 = 5 , 𝑦 = − 2 3
  • D 𝑥 = 9 2 9 , 𝑦 = − 2 3 2 9
  • E 𝑥 = − 9 1 1 , 𝑦 = 2 3 1 1

Q16:

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

  • A 𝑥 = 1 1 1 6 , 𝑦 = 2 3 3 2
  • B 𝑥 = 6 , 𝑦 = 1 4
  • C 𝑥 = 2 3 3 2 , 𝑦 = 1 1 1 6
  • D 𝑥 = − 1 1 1 6 , 𝑦 = − 2 3 3 2
  • E 𝑥 = 1 1 1 4 , 𝑦 = 2 3 2 8

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