Worksheet: Consistent and Inconsistent Systems of Equations

In this worksheet, we will practice determining the consistency of a system of equations.

Q1:

Determine the value of that makes the system of equations have infinitely many solutions.

Q2:

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

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

Q3:

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

Q4:

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.

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

Q5:

In the 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.

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

Q6:

In the augmented matrix 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 inconsistent.
• BThe matrix is consistent and there is no unique solution.
• CThe matrix is consistent and there is a unique solution.

Q7:

Suppose the coefficient matrix of a system, of equations with 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?

• A yes, no
• B No, there is no solution.
• C yes, yes

Q8:

Is there a value of that makes an augmented matrix of a consistent matrix? If yes, find the value of .

• Ayes,
• Bno

Q9:

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

Q10:

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

• Ano
• Byes

Q11:

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

• AIf , then there is exactly one solution. If and , then there are no solutions. If and , then there are infinitely many solutions.
• BIf , there are infinitely many solutions. If and , there are no solutions. If and , there will be a unique solution.
• CIf , then there is exactly one solution. If and , then there are no solutions. If and , then there are infinitely many solutions.
• DIf , then there is exactly one solution. If and , then there are no solutions. If and , then there are infinitely many solutions.
• EIf , then there is exactly one solution. If and , then there are no solutions. If and , then there are infinitely many solutions.

Q12:

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

• AIf there will be a unique solution. If and , there are no solutions. If and , then there are infinitely many solutions.
• BIf there are infinitely many solutions. If and , there are no solutions. If and , there will be a unique solution.
• CIf there are infinitely many solutions. If and , there are no solutions. If and , there will be a unique solution.
• DIf there will be a unique solution for any . If and , there are no solutions. If and , then there are infinitely many solutions.
• EIf there are infinitely many solutions. If and , there are no solutions. If and , there will be a unique solution.

Q13:

Find the solution for the system of equation

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