Lesson Worksheet: Cramer’s Rule Mathematics • 10th Grade

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


Consider the following simultaneous equations: 3π‘₯+2𝑦=8,βˆ’8π‘₯βˆ’9𝑦=2.

Write the determinant Δ.

  • A||3βˆ’8βˆ’8βˆ’2||
  • B||822βˆ’9||
  • C||βˆ’82βˆ’2βˆ’9||
  • D||32βˆ’8βˆ’9||
  • E||38βˆ’82||

Write the determinant Ξ”ο˜.

  • A||βˆ’82βˆ’2βˆ’9||
  • B||3βˆ’8βˆ’8βˆ’2||
  • C||32βˆ’8βˆ’9||
  • D||822βˆ’9||
  • E||38βˆ’82||

Write the determinant Ξ”.

  • A||38βˆ’82||
  • B||32βˆ’8βˆ’9||
  • C||822βˆ’9||
  • D||βˆ’82βˆ’2βˆ’9||
  • E||3βˆ’8βˆ’8βˆ’2||


Consider the following linear system of equations in matrix form: 9153π‘₯π‘¦οŸ=ο”βˆ’1βˆ’3.

Write the determinant Δ.

  • A||9βˆ’15βˆ’3||
  • B||1133||
  • C||9153||
  • D||βˆ’9βˆ’153||
  • E||βˆ’11βˆ’33||

Write the determinant Ξ”ο˜.

  • A||1133||
  • B||9153||
  • C||βˆ’11βˆ’33||
  • D||βˆ’9βˆ’153||
  • E||9βˆ’15βˆ’3||

Write the determinant Ξ”.

  • A||9βˆ’15βˆ’3||
  • B||9153||
  • C||βˆ’11βˆ’33||
  • D||1133||
  • E||βˆ’9βˆ’153||


Use determinants to solve the system βˆ’8π‘₯βˆ’4𝑦=βˆ’8,9π‘₯βˆ’6𝑦=βˆ’9.

  • Aπ‘₯=127, 𝑦=17
  • Bπ‘₯=17, 𝑦=127
  • Cπ‘₯=1, 𝑦=12
  • Dπ‘₯=βˆ’17, 𝑦=βˆ’127
  • Eπ‘₯=6, 𝑦=βˆ’10


Use determinants to solve the system π‘₯βˆ’5𝑦+3𝑧=5,3π‘₯βˆ’4𝑦+2𝑧=βˆ’5,βˆ’π‘₯+3π‘¦βˆ’2𝑧=βˆ’5.

  • Aπ‘₯=βˆ’53, 𝑦=203, 𝑧=403
  • Bπ‘₯=53, 𝑦=βˆ’203, 𝑧=βˆ’403
  • Cπ‘₯=βˆ’953, 𝑦=βˆ’703, 𝑧=1303
  • Dπ‘₯=53, 𝑦=βˆ’203, 𝑧=10


Use determinants to solve the system βˆ’9π‘₯=βˆ’8+8𝑦,6𝑦=7+3π‘₯.

  • Aπ‘₯=βˆ’439, 𝑦=2926
  • Bπ‘₯=8, 𝑦=βˆ’8
  • Cπ‘₯=2926, 𝑦=βˆ’439
  • Dπ‘₯=439, 𝑦=βˆ’2926
  • Eπ‘₯=βˆ’415, 𝑦=2910


According to Cramer’s rule and given that Ξ”=||3752||,Ξ”=||βˆ’53βˆ’25||,ο—ο˜ write the simultaneous equations of the system.

  • A7π‘₯βˆ’5𝑦=3,
  • Bβˆ’5π‘₯+7𝑦=3,
  • C7π‘₯βˆ’5𝑦=βˆ’3,
  • Dβˆ’5π‘₯+7𝑦=βˆ’3,
  • Eβˆ’5π‘₯+7𝑦=0,


Use determinants to solve the system 5π‘₯=βˆ’2π‘¦βˆ’5+3𝑧,βˆ’3π‘₯βˆ’π‘¦+1=2𝑧,2π‘¦βˆ’π‘§=βˆ’5π‘₯+3.

  • Aπ‘₯=4, 𝑦=3, 𝑧=313
  • Bπ‘₯=21, 𝑦=βˆ’56, 𝑧=βˆ’4
  • Cπ‘₯=βˆ’21, 𝑦=56, 𝑧=4
  • Dπ‘₯=2111, 𝑦=βˆ’5611, 𝑧=βˆ’811
  • Eπ‘₯=βˆ’27, 𝑦=29, 𝑧=βˆ’26


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

  • AYes
  • BNo


Noah is solving simultaneous equations using Cramer’s rule. He writes down the following: Ξ”=1234βˆ’3βˆ’221βˆ’4,Ξ”=213βˆ’34βˆ’2βˆ’12βˆ’4,Ξ”=221βˆ’3βˆ’34βˆ’112.ο—ο˜ο™

What does he write down for Ξ”?

  • Aο˜βˆ’11βˆ’4βˆ’3βˆ’34223
  • B223βˆ’3βˆ’3βˆ’2βˆ’11βˆ’4
  • C322βˆ’3βˆ’34βˆ’41βˆ’1
  • Dο˜βˆ’11βˆ’4βˆ’34βˆ’2223
  • E221βˆ’3βˆ’34βˆ’112


Solve, using Cramer's rule, the simultaneous equations |||βˆ’1π‘§βˆ’4𝑦|||=23,|||2π‘¦βˆ’5π‘₯|||=13,||3π‘₯5𝑧||=51.

  • Aπ‘₯=70253, 𝑦=βˆ’40353, 𝑧=7
  • Bπ‘₯=66347, 𝑦=5, 𝑧=βˆ’37147
  • Cπ‘₯=βˆ’6, 𝑦=5, 𝑧=7
  • Dπ‘₯=βˆ’7, 𝑦=5, 𝑧=9
  • Eπ‘₯=βˆ’6, 𝑦=7, 𝑧=6

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