Worksheet: Homogeneous Systems of Linear Equations

In this worksheet, we will practice finding the solution of a homogeneous system of linear equations and expressing the general solution in case of an infinite number of solutions.

Q1:

Which of the following systems of equations is homogeneous?

  • Aβˆ’6π‘₯+2𝑦=0, 7π‘₯βˆ’8𝑦=0
  • B5π‘₯βˆ’7𝑦=4, βˆ’2π‘₯βˆ’9𝑦=15
  • C5π‘₯βˆ’7𝑦=4, βˆ’2π‘₯βˆ’9𝑦=0
  • Dβˆ’6π‘₯+2𝑦=0, 7π‘₯βˆ’8𝑦=15

Q2:

Find the value of π‘˜ that would make the equations 4π‘₯+9𝑦+5𝑧=0, 16π‘₯+36𝑦+π‘˜π‘§=0, and 9π‘₯βˆ’8π‘¦βˆ’3𝑧=0 have a solution other than zero.

Q3:

Find all the solutions to the system of equations 10π‘₯βˆ’6𝑦=0, βˆ’15π‘₯+9𝑦=0.

  • AThe system has no solution.
  • Bπ‘₯=3𝑑, 𝑦=5𝑑, where 𝑑 is an integer
  • Cπ‘₯=5, 𝑦=3
  • DThe system only has a trivial solution: π‘₯=0, 𝑦=0.
  • Eπ‘₯=5𝑑, 𝑦=3𝑑, where 𝑑 is an integer

Q4:

Find all the solutions of the system of equations 2π‘₯βˆ’4𝑦+6𝑧=0, βˆ’5π‘₯+3𝑦+𝑧=0, 6π‘₯+5𝑦+4𝑧=0.

  • AThe system only has a trivial solution π‘₯=0, 𝑦=0, 𝑧=0.
  • Bπ‘₯=𝑑, 𝑦=𝑑, 𝑧=βˆ’2𝑑, where 𝑑 is an integer
  • Cπ‘₯=1, 𝑦=1, 𝑧=βˆ’2
  • DThe system has no solution.
  • Eπ‘₯=𝑑, 𝑦=βˆ’2𝑑, 𝑧=𝑑, where 𝑑 is an integer

Q5:

Find all the solutions of the system of equations 2π‘₯+4𝑦+6𝑧=0, 5π‘₯+3𝑦+𝑧=0, 6π‘₯+5𝑦+4𝑧=0.

  • AThe system has no solution.
  • Bπ‘₯=1, 𝑦=1, 𝑧=βˆ’2
  • Cπ‘₯=𝑑, 𝑦=𝑑, 𝑧=βˆ’2𝑑, where 𝑑 is an integer
  • DThe system only has a trivial solution π‘₯=0, 𝑦=0, 𝑧=0.
  • Eπ‘₯=𝑑, 𝑦=βˆ’2𝑑, 𝑧=𝑑, where 𝑑 is an integer

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.