Free Practice Quiz Question List

3 Systems of Linear Equations Online Quiz Questions

Use this free practice quiz with 20 questions to review 3 Systems of Linear Equations, test your knowledge, and prepare for your next test or exam.

20 questions

Choose one

1 point

Starting with the augmented matrix $\left[\begin{array}{cc|c}1&2&5\\3&4&11\end{array}\right]$ , what matrix results from the row operation $R_2\leftarrow R_2-3R_1$ ?

A
$\left[\begin{array}{cc|c}1&2&5\\0&4&-4\end{array}\right]$
B
$\left[\begin{array}{cc|c}1&2&5\\0&-2&-4\end{array}\right]$
C
$\left[\begin{array}{cc|c}1&2&5\\3&1&6\end{array}\right]$
D
$\left[\begin{array}{cc|c}1&2&5\\0&-2&4\end{array}\right]$

Choose one

1 point

A row-reduced augmented matrix contains the row $[0\ 0\ 0\mid 4]$ . What is the solution type of the corresponding system?

A
No solution
B
Exactly one solution
C
Infinitely many solutions
D
A solution only when all variables are positive

Choose one

1 point

What is the solution type of the system represented by $\left[\begin{array}{ccc|c}1&0&2&5\\0&1&-3&1\\0&0&0&0\end{array}\right]$ ?

A
No solution because one variable is free
B
A unique solution because there are two pivots
C
Infinitely many solutions because the system is consistent and one variable is free
D
Infinitely many solutions because the augmented column has a pivot

Choose all

1 point

Which of the following are elementary row operations that preserve the solution set? Select all correct choices.

A
Interchanging two rows
B
Multiplying a row by zero
C
Multiplying a row by a nonzero constant
D
Deleting a row that appears redundant

Choose all

1 point

Which statements correctly describe pivots, rank, and solution types? Select all correct choices.

A
A pivot in every variable column implies a unique solution, provided the system is consistent.
B
A consistent system with a nonpivot variable column has infinitely many solutions.
C
A pivot in the augmented column guarantees a unique solution.
D
The number of pivots in the coefficient matrix is the rank of that matrix.

True or false

1 point

True or false: Two row-equivalent augmented matrices represent systems with exactly the same solution set.

A
True
B
False

True or false

1 point

True or false: Gaussian elimination and Gauss–Jordan elimination always stop at the same matrix form.

A
True
B
False

Written response

1 point

A consistent system in three variables has a coefficient matrix with rank $2$ . How many free variables does it have?

Fill in the blank

1 point

In the matrix representation of a system, the matrix containing only the variable coefficients is the , while combining it with the constants column produces the .

Fill in the blank

1 point

A variable in a nonpivot column is a ; when describing infinitely many solutions, it is assigned a such as $t$ .

Open ended

1 point

For the augmented matrix $\left[\begin{array}{ccc|c}1&0&2&5\\0&1&-3&1\\0&0&0&0\end{array}\right]$ , determine whether the system is consistent and describe its complete solution set in parametric form.

Choose one

1 point

A laboratory needs $10$ liters of a solution that is $30\%$ acid. It mixes a $20\%$ solution, $x$ liters, with a $50\%$ solution, $y$ liters. What amounts satisfy both the total-volume and acid-content equations?

A
$x=5$ liters and $y=5$ liters
B
$x=\frac{20}{3}$ liters and $y=\frac{10}{3}$ liters
C
$x=\frac{10}{3}$ liters and $y=\frac{20}{3}$ liters
D
$x=8$ liters and $y=2$ liters

Choose one

1 point

In the mixture model with $10$ liters of final solution at $30\%$ acid, which equation represents the total amount of acid?

A
$x+y=3$
B
$0.20x+0.50y=10$
C
$0.20x+0.50y=3$
D
$0.30x+0.30y=10$

Choose one

1 point

Which elementary row operation always preserves the solution set of a linear system?

A
Multiply a row by zero
B
Multiply a row by a nonzero constant
C
Multiply a row by a variable that may equal zero
D
Delete a row without checking whether it is redundant

Choose one

1 point

What can be concluded from an augmented matrix that contains the row $\left[\begin{array}{ccc|c}0&0&0&4\end{array}\right]$ ?

A
The system has a unique solution
B
The system has infinitely many solutions
C
The system has no solution
D
The system has exactly two solutions

Choose one

1 point

A consistent system in three variables has an RREF with pivots in the $x$ - and $y$ -columns but no pivot in the $z$ -column. What type of solution set does it have?

A
The system has a unique solution because it has two pivot rows
B
The system has infinitely many solutions because one variable is free
C
The system has no solution because one variable is free
D
The system has exactly two solutions because there are two pivot variables

True or false

1 point

True or false: Gaussian elimination must continue until reduced row-echelon form, whereas Gauss–Jordan elimination stops at ordinary row-echelon form.

A
True
B
False

Written response

1 point

A laboratory mixes a 20% acid solution with a 50% acid solution to obtain 10 liters of a 30% acid solution. If $x$ is the amount of 20% solution and $y$ is the amount of 50% solution, what is $y$ in liters? Enter the exact value; a fraction is allowed.

Written response

1 point

Solve the system and determine the exact value of $z$ :

\begin{cases} x+y+z=6,\\ x-y+z=4,\\ 2x+y-z=5. \end{cases}

Written response

1 point

Solve the system and determine the exact value of $y$ :

\begin{cases} x+2y=11,\\ 3x-y=5. \end{cases}