Worksheet: Cramer's Rule

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=||9822||,Δ=||9932||.

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+2𝑦2𝑧1=0,3𝑥+3𝑦3𝑧+3=0,2𝑥+4𝑦5𝑧+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=4𝑦3𝑧,2𝑥2𝑦+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 8𝑥4𝑦=8,9𝑥6𝑦=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+8𝑦1=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=2𝑦5+3𝑧,3𝑥𝑦+1=2𝑧,2𝑦𝑧=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 |||1𝑧4𝑦|||=23,|||2𝑦5𝑥|||=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 𝑥5𝑦+3𝑧=5,3𝑥4𝑦+2𝑧=5,𝑥+3𝑦2𝑧=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 𝑎𝑑𝑏𝑐=1. The solution vector is 𝑥𝑦=𝑒𝑏𝑓𝑑.det

  • AFalse
  • BTrue

Q14:

Use determinants to solve the system 2𝑥2𝑦+4𝑧+2=0,5𝑥4𝑦+2𝑧5=0,2𝑥+3𝑦+𝑧2=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+𝑦2=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=22𝑦,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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