Worksheet: Solution Cases of System of Linear Equations

In this worksheet, we will practice determining whether a linear system of equations has a unique solution, no solution, or an infinite number of solutions.

Q1:

Determine the number of solutions to the system of equations 6𝑥2𝑦+9𝑧=4,24𝑥8𝑦+36𝑧=22,12𝑥4𝑦+18𝑧=10.

  • AThere are no solutions.
  • BThere are finitely many solutions, not including 𝑥=0,𝑦=0,𝑧=0.
  • CThere are infinitely many solutions.
  • DThe only solution is 𝑥=0,𝑦=0,𝑧=0.

Q2:

Determine the number of solutions to the following system: 4801301101417𝑥𝑦𝑧=000.

  • AThere are infinitely many solutions.
  • BThere are finitely many solutions, not including 𝑥=0,𝑦=0,𝑧=0.
  • CThe only solution is 𝑥=0,𝑦=0,𝑧=0.
  • DThere are no solutions.

Q3:

Determine the number of solutions to the system of equations 4𝑥4𝑦=0,8𝑥2𝑧=0,5𝑦7𝑧=0.

  • AThe only solution is 𝑥=0,𝑦=0,𝑧=0.
  • BThere are finitely many solutions, not including 𝑥=0,𝑦=0,𝑧=0.
  • CThere are no solutions.
  • DThere are infinitely many solutions.

Q4:

Determine the number of solutions to the system of equations 2𝑥+2𝑦=0,3𝑥3𝑧=0,7𝑦+4𝑧=0.

  • AThe only solution is 𝑥=0,𝑦=0,𝑧=0.
  • BThere are finitely many solutions, not including 𝑥=0,𝑦=0,𝑧=0.
  • CThere are no solutions.
  • DThere are infinitely many solutions.

Q5:

Determine the number of solutions to the system of equations 9𝑥3𝑦+8𝑧=4,45𝑥15𝑦+40𝑧=20,18𝑥6𝑦+16𝑧=8.

  • AThere are infinitely many solutions.
  • BThere are finitely many solutions, not including 𝑥=0,𝑦=0,𝑧=0.
  • CThere are no solutions.
  • DThe only solution is 𝑥=0,𝑦=0,𝑧=0.

Q6:

Determine the number of solutions to the following system: 1413070210320𝑥𝑦𝑧=000.

  • AThere are infinitely many solutions.
  • BThere are finitely many solutions, not including 𝑥=0,𝑦=0,𝑧=0.
  • CThe only solution is 𝑥=0,𝑦=0,𝑧=0.
  • DThere are no solutions.

Q7:

Determine the number of solutions to the following system: 50231506910046𝑥𝑦𝑧=123619.

  • AThe only solution is 𝑥=0,𝑦=0,𝑧=0.
  • BThere are finitely many solutions, not including 𝑥=0,𝑦=0,𝑧=0.
  • CThere are no solutions.
  • DThere are infinitely many solutions.

Q8:

Find such that 13246 is the augmented matrix of a consistent system of equations.

  • AThere is no value of that makes this the augmented matrix of a consistent system.
  • Bany real number

Q9:

If there is a unique solution to a system of linear equations, which of the following must be true of the columns in the augmented matrix?

  • AThe last one must not be a pivot, and the ones to the left must each be pivot columns.
  • BThe last one must be a pivot, and the ones to the left must each be pivot columns.
  • CThe last one must not be a pivot, and the ones to the left must not each be pivot columns.
  • DThe last one must be a pivot, and the ones to the left must not each be pivot columns.

Q10:

In the given augmented matrix, * denotes an arbitrary number and ■ denotes an arbitrary nonzero number. Determine whether the given augmented matrix is consistent and, if it is consistent, whether its solution is unique: 00000000.

  • AThe matrix is consistent and its solution is unique.
  • BThe matrix is inconsistent.
  • CThe matrix is consistent and its solution is not unique.

Q11:

In the augmented matrix 000, denotes an arbitrary number and denotes a nonzero number. Determine whether the given augmented matrix is consistent. If it is consistent, is the solution unique?

  • AThe matrix is consistent and there is a unique solution.
  • BThe matrix is inconsistent.
  • CThe matrix is consistent and there is no unique solution.

Q12:

In the augmented matrix 00000000000, denotes an arbitrary number and denotes an arbitrary nonzero number. Determine whether the given augmented matrix is consistent and, if it is consistent, whether its solution is unique.

  • AThe matrix is consistent and its solution is not unique.
  • BThe matrix is consistent and its solution is unique.
  • CThe matrix is inconsistent.

Q13:

Suppose the coefficient matrix of a system, of 𝑛 equations with 𝑛 variables, has the property that every column is a pivot column. Does the system of equations have a solution? If so, must the solution be unique?

  • Ayes, yes
  • BNo, there is no solution.
  • Cyes, no

Q14:

State all values of that make the augmented matrix 114312 consistent.

  • AAny value of except 3
  • BAny value of
  • C = 3
  • D > 0
  • EThere is no which makes the augmented matrix consistent.

Q15:

Find conditions on and 𝑘 for the following augmented matrix to have no solution, a unique solution, and infinitely many solutions: 1222𝑘.

  • AIf =1, then there is exactly one solution. If 1 and 𝑘1, then there are no solutions. If =1 and 𝑘=1, then there are infinitely many solutions.
  • BIf =4, then there is exactly one solution. If 4 and 𝑘4, then there are no solutions. If =4 and 𝑘=4, then there are infinitely many solutions.
  • CIf 4, then there is exactly one solution. If =4 and 𝑘4, then there are no solutions. If =4 and 𝑘=4, then there are infinitely many solutions.
  • DIf =2, then there is exactly one solution. If 2 and 𝑘2, then there are no solutions. If =2 and 𝑘=2, then there are infinitely many solutions.
  • EIf 4, there are infinitely many solutions. If =4 and 𝑘4, there are no solutions. If =4 and 𝑘=4, there will be a unique solution.

Q16:

Find the value of for which the augmented matrix 24367 is inconsistent.

Q17:

Suppose a system of linear equations has a 2×4 augmented matrix and the last column is a pivot column. Could the system of linear equations be consistent?

  • Ano
  • Byes

Q18:

Find conditions on and 𝑘 for the augmented matrix 1224𝑘 to have no solution, a unique solution, and infinitely many solutions.

  • AIf 2, there will be a unique solution for any 𝑘. If =2 and 𝑘4, there are no solutions. If =2 and 𝑘=4, then there are infinitely many solutions.
  • BIf 4, there will be a unique solution. If =4 and 𝑘2, there are no solutions. If =4 and 𝑘=2, then there are infinitely many solutions.
  • CIf 4, there are infinitely many solutions. If =4 and 𝑘2, there are no solutions. If =4 and 𝑘=2, there will be a unique solution.
  • DIf 1, there are infinitely many solutions. If =1 and 𝑘2, there are no solutions. If =1 and 𝑘=2, there will be a unique solution.
  • EIf 2, there are infinitely many solutions. If =2 and 𝑘4, there are no solutions. If =2 and 𝑘=4, there will be a unique solution.

Q19:

True or False: If 𝐴 is a matrix and 𝑏 is not the zero vector, then 𝐴𝑥=𝑏 has a solution if and only if 𝑏 can be written as a linear combination of the columns of 𝐴.

  • ATrue
  • BFalse

Q20:

Find the solution for the system of equation 𝑥+2𝑦+𝑧𝑤=2,𝑥𝑦+𝑧+𝑤=0,2𝑥+𝑦𝑧=1,4𝑥+2𝑦+𝑧=3.

If there is no solution, state the reason.

  • AThe system of equations is not consistent.
  • BThe system of equations is consistent and its solution is not unique.
  • CThe system of equations is consistent and its solution is unique.

Q21:

Determine the value of 𝑘 that makes the system of equations 7𝑥+5𝑦=7,35𝑥+25𝑦=𝑘 have infinitely many solutions.

Q22:

Suppose 𝐴 is matrix in × and v is a vector in and consider this statement: there exists a solution x to the equation 𝐴=xv. Which of the following is true?

  • AThe statement is always true.
  • BIf 𝐴 is nonsingular, then the statement is true.
  • CIf the statement is true, then 𝐴 is nonsingular.
  • DIf 𝐴 is nonsingular, then the statement is false.
  • EThe statement is true if and only if 𝐴 is nonsingular.

Q23:

If 𝑏0 and 𝐴 is an invertible 𝑛×𝑛 matrix, can the solution set of 𝐴𝑥=𝑏 be a plane through the origin?

  • ANo
  • BYes

Q24:

Find the set of values of 𝑘 for which the simultaneous equations 9𝑥9𝑦5𝑧=6,2𝑥+3𝑦+7𝑧=4,3𝑥4𝑦+𝑘𝑧=7, have at least one solution.

  • A 1 4 8 4 5
  • B 2 2 4 5
  • C 2 2 4 5
  • D 4 3 6 4 5
  • E 1 4 8 4 5

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