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?

  • Ano
  • Byes

Q2:

William is solving simultaneous equations using Cramer’s rule. He writes down Δ = | | − 9 − 8 2 − 2 | | , Δ = | | 9 − 9 − 3 2 | | .

What does he write down for Δ ?

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

Q3:

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

  • A 𝑥 = 1 9 , 𝑦 = − 2 1 , 𝑧 = − 1 3
  • 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 = 4 , 𝑦 = − 4 , 𝑧 = 3 2

Q4:

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

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

Q5:

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

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

Q6:

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

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

Q7:

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

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

Q8:

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

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

Q9:

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

  • A 𝑥 = 6 6 3 4 7 , 𝑦 = 5 , 𝑧 = − 3 7 1 4 7
  • B 𝑥 = − 7 , 𝑦 = 5 , 𝑧 = 9
  • C 𝑥 = 7 0 2 5 3 , 𝑦 = − 4 0 3 5 3 , 𝑧 = 7
  • D 𝑥 = − 6 , 𝑦 = 5 , 𝑧 = 7
  • 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 22 LE and a crayon costs 1,078 LE.
  • BA sketchbook costs 441 LE and a crayon costs 1,078 LE.
  • CA sketchbook costs 441 LE and a crayon costs 9 LE.
  • DA sketchbook costs 22 LE and a crayon costs 9 LE.
  • EA sketchbook costs 9 LE and a crayon costs 2.70 LE.

Q11:

Use Cramer’s rule to solve the simultaneous equations | | | − 3 𝑦 2 𝑥 | | | = − 9 , | | | 1 𝑧 1 𝑦 | | | = − 1 , | | 4 𝑥 4 𝑧 | | = 2 .

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

Q12:

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

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

Q13:

True or False: Suppose that 𝑎 𝑥 + 𝑏 𝑦 = 𝑒 , 𝑐 𝑥 + 𝑑 𝑦 = 𝑓 is a system of equations with 𝑎 𝑑 − 𝑏 𝑐 = 1 . The solution vector is 𝑥 𝑦 = 𝑒 𝑏 𝑓 𝑑 . d e t

  • AFalse
  • BTrue

Q14:

Given that 9 𝑦 + 5 𝑥 = 8 and 1 𝑦 + 1 𝑥 = 1 , determine the values of 𝑥 and 𝑦 .

  • A 𝑥 = 4 , 𝑦 = 3 4
  • B 𝑦 = 4 , 𝑥 = 4 3
  • C 𝑦 = 4 , 𝑥 = 3 4
  • D 𝑥 = 4 , 𝑦 = 4 3

Q15:

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

  • A 𝑥 = − 1 1 9 , 𝑦 = − 4 0 5 7 , 𝑧 = 1 0 1 9
  • 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 = 9 , 𝑦 = − 1 , 𝑧 = 7 2

Q16:

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

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

Q17:

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

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

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