Question 1

Choose the one alternative that best completes the statement or answers the question.
For the given augmented matrix, determine the number of solutions to the corresponding system of
equations.
-$$\left[ \begin{array} { l l l | r } 
1 & 0 & 0 & - 3 \
0 & 1 & 0 & - 9 \
0 & 0 & 1 & 0
\end{array} \right]$$
A) No solution
B) One solution
C) Infinitely many solutions

Accepted Answer

The augmented matrix is in reduced row echelon form with each variable leading in a separate equation, indicating a unique solution for each variable. This corresponds to a single, unique solution for the system of equations.

Question 2

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-If a system of linear equations has infinitely many solutions, then the equations are said to be________

Accepted Answer

The answer of Write the word or phrase that best...

Question 3

The solution set to a system of dependent equations is given. Write the specific solution corresponding to
the given value of z.
-$\{ ( 4 z - 7 , z - 4 , z ) \mid z$ is any real number $\}$ $z = - 5$
A) $( - 27,1 , - 5 )$
B) $( - 27 , - 9 , - 5 )$
C) $( - 13,1 , - 5 )$
D) $( - 13 , - 9 , - 5 )$

Accepted Answer

Substituting $z = -5$ into each expression gives: $4(-5) - 7 = -20 - 7 = -27$, $(-5) - 4 = -9$, and $z = -5$, resulting in the solution $(-27, -9, -5)$.

Question 4

Determine the solution set for the system represented by the augmented matrix.
-$\left[ \begin{array} { r r r | r } 1 & 0 & - 7 & 4 \ 0 & 1 & 6 & 1 \ 0 & 0 & 0 & 1 \end{array} \right]$
A) $\{ ( - 7,6,0 ) \}$
B) $\{ ( 11 , - 5,1 ) \}$
C) $\{ ( 4,1,0 ) \}$
D){ }

Accepted Answer

The augmented matrix shows that the system of equations is inconsistent due to the row $[0 \ 0 \ 0 \ | \ 1]$, which implies $0=1$, an impossibility. Therefore, there is no solution.

Question 5

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
-Assume that traffic flows freely through intersections A, B, and C. The values $$x _ { 1 } , x _ { 2 } , x _ { 3 }$$ , and all other
numbers in the figure represent flow rates in vehicles per hour.  a. Write an equation representing equal flow into and out of intersection A.
b. Write an equation representing equal flow into and out of intersection B.
c. Write an equation representing equal flow into and out of intersection C.
d. Write the system of equations from parts (a)-(c) in standard form.
e. Write the reduced row-echelon form of the augmented matrix representing the system of equations from
part (d).
f. If the flow rate between intersections A and C is 120 vehicles per hour, determine the flow rates $x _ { 1 }$ and $x _ { 2 }$ g. If the flow rate between intersections A and C is between 150 and 240 vehicles per hour, inclusive,
determine the flow rates $x _ { 1 } \text { and } x _ { 2 }$

Accepted Answer

The answer of Write the word or phrase that best...

Question 6

The solution set to a system of dependent equations is given. Write three ordered triples that are
solutions to the system. Answers may vary.
-$\left\{ \left( \frac { 6 - 4 y - 15 z } { 3 } , y , z \right) \mid y \right.$ and $z$ are any real numbers $\}$
A) $( 5,0,0 ) , ( 2,3,0 ) , ( - 3,0,1 )$
B) $( 2,0,0 ) , ( 2,3,0 ) , ( 1,0,1 )$
C) $( 5,0,0 ) , ( - 2,3,0 ) , ( 1,0,1 )$
D) $( 2,0,0 ) , ( - 2,3,0 ) , ( - 3,0,1 )$

Accepted Answer

The answer of The solution set to a system of...

Question 7

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-If a system of linear equations has no solution, then the system is said to be________ .

Accepted Answer

The answer of Write the word or phrase that best...

Question 8

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
-$$\begin{array} { l } 
- 6 x + 15 y = 15 \
2 x - 5 y = - 5
\end{array}$$
A) $\{ ( 15 , - 5 ) \}$
B) $\left\{ \left( \frac { 5 } { 2 } x - \frac { 5 } { 2 } , y \right) \mid y \right.$ is any real number
C) $\{ ( - 9 , - 5 ) \}$
D){}

Accepted Answer

The system of equations is consistent and dependent, meaning it has infinitely many solutions. The second equation is a multiple of the first, indicating that both equations represent the same line. Therefore, the solution set involves a parameter (in this case, $y$), indicating that for any value of $y$, there's a corresponding $x$ value that satisfies both equations.

Question 9

Choose the one alternative that best completes the statement or answers the question.
For the given augmented matrix, determine the number of solutions to the corresponding system of
equations.
-$$\left[ \begin{array} { r r r | r } 
1 & 0 & 7 & - 2 \
0 & 1 & - 8 & 1 \
0 & 0 & 0 & 0
\end{array} \right]$$
A) One solution
B) Infinitely many solutions
C) No solution

Accepted Answer

The given augmented matrix shows that the system of equations has variables that can take on an infinite number of values because the last row [0 0 0 | 0] indicates that there are more variables than independent equations. This typically results in a system with infinitely many solutions.

Question 10

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
-$$\begin{array} { l } 
8 x + 10 y = - 17 \
4 x + 5 y = - 9
\end{array}$$
A) $\{ ( - 7,7 ) \}$
B) $\{ ( - 17 , - 9 ) \}$
C) $\left\{ \left( - \frac { 5 } { 4 } x - \frac { 9 } { 4 } , y \right) \mid y \right.$ is any real number $\}$
D) { }

Accepted Answer

The answer of Solve the system using Gaussian elimination or...

Question 11

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
-$\begin{array} { l } 
3 x + 2 y + 7 z = 38 \
9 x + 6 y + 21 z = 114 \
0.6 x + 0.4 y + 1.4 z = 7.6
\end{array}$

A){ }
B) { ( x , y , z )|  3 x +2 y +7 z = 38 }
C) $\{ ( -1,-4,7 ) \}$
D) $\{ ( 1,4,-7) \}$

Accepted Answer

The answer of Solve the system using Gaussian elimination or...

Question 12

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
-$$\begin{array} { l } 
- 4 x - 6 y + 5 z = 13 \
8 x + 4 y - 7 z = - 7 \
4 x - 2 y - 2 z = - 15
\end{array}$$
A) $\{ ( 8,4,5 ) \}$
B) $\{ ( 8,5,4 ) \}$
C) { }
D) $\{ ( 5,8,4 ) \}$

Accepted Answer

The system of equations is inconsistent, meaning there is no solution.

Question 13

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
-$$\begin{array} { l } 
- 2 x - y + 3 z = 13 \
x - 3 y - 3 z = - 4
\end{array}$$
A) $\{ ( - 1 , - 2,3 ) \}$
B) $\left\{ \left( \frac { 12 } { 7 } z - \frac { 43 } { 7 } , - \frac { 3 } { 7 } z - \frac { 5 } { 7 } , z \right) \mid z \right.$ is any real number $\}$
C){}
D) $\{ ( 1,2 , - 3 ) \}$

Accepted Answer

This system of equations has two equations and three unknowns, which means it's underdetermined and will have infinitely many solutions, typically parameterized by one of the variables. By applying Gaussian elimination, we can simplify the system to express two of the variables in terms of the third. The correct answer reflects this, showing x and y in terms of z, indicating that for any real value of z, there's a corresponding solution for x and y.

Question 14

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
- a. Assume that traffic flows at a rate of 215 vehicles per hour on the stretch of road between intersections $\mathrm { D }$ and $\mathrm { A }$. Find the flow rates $x _ { 1 } , x _ { 2 }$, and $x _ { 3 }$.
b. If traffic flows at a rate of between 195 and 270 vehicles per hour inclusive between intersections $\mathrm { D }$ and $\mathrm { A }$, find the flow rates $x _ { 1 } , x _ { 2 }$, and $x _ { 3 }$.
A) a. $x _ { 1 } = 65 , x _ { 2 } = 195$, and $x _ { 3 } = 170$;
b. $40 \leq x _ { 1 } \leq 115,180 \leq x _ { 2 } \leq 255$, and $155 \leq x _ { 3 } \leq 230$
B) a. $x _ { 1 } = 65 , x _ { 2 } = 195$, and $x _ { 3 } = 170$;
b. $35 \leq x _ { 1 } \leq 110,185 \leq x _ { 2 } \leq 260$, and $145 \leq x _ { 3 } \leq 220$
C) a. $x _ { 1 } = 55 , x _ { 2 } = 205$, and $x _ { 3 } = 165$;
b. $40 \leq x _ { 1 } \leq 115,180 \leq x _ { 2 } \leq 255$, and $155 \leq x _ { 3 } \leq 230$
D) a. $x _ { 1 } = 55 , x _ { 2 } = 205$, and $x _ { 3 } = 165$;
b. $35 \leq x _ { 1 } \leq 110,185 \leq x _ { 2 } \leq 260$, and $145 \leq x _ { 3 } \leq 220$

Accepted Answer

The answer of Write the word or phrase that best...

Question 15

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-True or false? A system of linear equations in three variables may have infinitely many solutions.

Accepted Answer

A system of linear equations in three variables can have infinitely many solutions if the equations are dependent, meaning they represent the same plane or intersecting planes that overlap along a line.

Question 16

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-If the ________of a matrix is $$p \times q$$ , then p represents the number of________ and q represents the number of .

Accepted Answer

The answer of Write the word or phrase that best...

Question 17

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
-An accountant checks the reported earnings for a concert venue for three different performers against the number of tickets sold. $\begin{array}{ccccc}
\text { Performer } & \begin{array}{c}
\text { Children } \
\text { Tickets }
\end{array} & \begin{array}{c}
\text { Student } \
\text { Tickets }
\end{array} & \begin{array}{c}
\text { General } \
\text { Admission }
\end{array} & \begin{array}{c}
\text { Total } \
\text { Revenue }
\end{array} \
\mathbf{1} & 1,000 & 250 & 1,000 & \$ 24,750 \
\mathbf{2} & 400 & 300 & 300 & \$ 10,700 \
\mathbf{3} & 400 & 300 & 300 & \$ 8,700
\end{array}$
 Let $x , y$, and $z$ represent the cost for children tickets, student tickets, and general admission tickets, respectively. Set up an augmented matrix for the system and solve for $x , y$, and $z$. Explain what the accountant knows about the reported earnings.

A){  } ; The system of equations reduces to a contradiction. There are no values for $x , y$, and $z$ that can simultaneously meet the conditions of this problem.
B) $\{ ( 7,7,16 ) \}$; There were $\$ 7$ children's tickets, $\$ 7$ student tickets, and $\$ 16$ general admission tickets sold.
C) $\{ ( 8,8,17 ) \}$; There were $\$ 8$ children's tickets, $\$ 8$ student tickets, and $\$ 17$ general admission tickets sold.
D) $\{ ( 6,6,15 ) \}$; There were $\$ 6$ children's tickets, $\$ 6$ student tickets, and $\$ 15$ general admission tickets sold.

Accepted Answer

The answer of Write the word or phrase that best...

Question 18

Determine the solution set for the system represented by the augmented matrix.
-$\left[ \begin{array} { r r r | r } 1 & 0 & - 4 & 7 \ 0 & 1 & 8 & - 7 \ 0 & 0 & 0 & 0 \end{array} \right]$
A) $\{ ( 7 - 4 z , - 7 + 8 z , z ) \mid z$ is any real number $\}$
B){ }
C) $\{ ( 7 , - 7,0 ) \}$
D) $\{ ( 7 + 4 z , - 7 - 8 z , z ) \mid z$ is any real number $\}$

Accepted Answer

The answer of Determine the solution set for the system...

Question 19

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
-$\begin{array} { l } 
- 3 x - 3 y - 3 z = 30 \
- 9 x - 9 y - 9 z = 90 \
- 1.5 x - 1.5 y - 1.5 z = 15
\end{array}$

A){ }
B) $\{ ( 2,2,6 ) \}$
C) $\{ ( -2,-2,-6 ) \}$
D) { ( x , y , z )| - 3 x - 3 y - 3 z = 30 }

Accepted Answer

All equations in the system are multiples of each other, meaning they represent the same plane in three-dimensional space. Therefore, there are infinitely many solutions along this plane, not a single unique solution.

Question 20

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
-$$\begin{array} { l } 
- 3 x - 7 y + 7 z = 71 \
- 2 x + 7 y - 3 z = - 22 \
- 3 x - 9 y + 9 z = 87
\end{array}$$
A) $\{ ( - 5 , - 2,6 ) \}$
B){ }
C) $\{ ( 5,2,6 ) \}$
D) $\{ ( - 5,2,6 ) \}$

Accepted Answer

Using Gaussian elimination, the system reduces to a unique solution: $x = -5$, $y = -2$, $z = 6$.

Quiz 6: Matrices and Determinants and Applications

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If a system of linear equations has infinitely many solutions, then the equations are said to be________

The solution set to a system of dependent equations is given. Write the specific solution corresponding to the given value of z. - $\{ ( 4 z - 7 , z - 4 , z ) \mid z$ is any real number $\}$ $z = - 5$

Determine the solution set for the system represented by the augmented matrix. - $\left[ \begin{array} { r r r | r } 1 & 0 & - 7 & 4 \\ 0 & 1 & 6 & 1 \\ 0 & 0 & 0 & 1 \end{array} \right]$

The solution set to a system of dependent equations is given. Write three ordered triples that are solutions to the system. Answers may vary. - $\left\{ \left( \frac { 6 - 4 y - 15 z } { 3 } , y , z \right) \mid y \right.$ and $z$ are any real numbers $\}$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If a system of linear equations has no solution, then the system is said to be________ .

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } - 6 x + 15 y = 15 \\2 x - 5 y = - 5\end{array}$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } 8 x + 10 y = - 17 \\4 x + 5 y = - 9\end{array}$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } 3 x + 2 y + 7 z = 38 \\9 x + 6 y + 21 z = 114 \\0.6 x + 0.4 y + 1.4 z = 7.6\end{array}$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } - 4 x - 6 y + 5 z = 13 \\8 x + 4 y - 7 z = - 7 \\4 x - 2 y - 2 z = - 15\end{array}$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } - 2 x - y + 3 z = 13 \\x - 3 y - 3 z = - 4\end{array}$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -True or false? A system of linear equations in three variables may have infinitely many solutions.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If the of a matrix is $p \times q$ , then p represents the number of and q represents the number of .

Determine the solution set for the system represented by the augmented matrix. - $\left[ \begin{array} { r r r | r } 1 & 0 & - 4 & 7 \\ 0 & 1 & 8 & - 7 \\ 0 & 0 & 0 & 0 \end{array} \right]$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } - 3 x - 3 y - 3 z = 30 \\- 9 x - 9 y - 9 z = 90 \\- 1.5 x - 1.5 y - 1.5 z = 15\end{array}$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } - 3 x - 7 y + 7 z = 71 \\- 2 x + 7 y - 3 z = - 22 \\- 3 x - 9 y + 9 z = 87\end{array}$

Quiz 6: Matrices and Determinants and Applications

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If a system of linear equations has infinitely many solutions, then the equations are said to be________

The solution set to a system of dependent equations is given. Write the specific solution corresponding to the given value of z. - {(4z−7,z−4,z) ∣z\{ ( 4 z - 7 , z - 4 , z ) \mid z{(4z−7,z−4,z) ∣z is any real number }\}} z=−5z = - 5z=−5

Determine the solution set for the system represented by the augmented matrix. - [10−7401610001]\left[ \begin{array} { r r r | r } 1 & 0 & - 7 & 4 \\ 0 & 1 & 6 & 1 \\ 0 & 0 & 0 & 1 \end{array} \right]​100​010​−760​411​​

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If a system of linear equations has no solution, then the system is said to be________ .

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - −6x+15y=152x−5y=−5\begin{array} { l } - 6 x + 15 y = 15 \\2 x - 5 y = - 5\end{array}−6x+15y=152x−5y=−5​

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - 8x+10y=−174x+5y=−9\begin{array} { l } 8 x + 10 y = - 17 \\4 x + 5 y = - 9\end{array}8x+10y=−174x+5y=−9​

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - 3x+2y+7z=389x+6y+21z=1140.6x+0.4y+1.4z=7.6\begin{array} { l } 3 x + 2 y + 7 z = 38 \\9 x + 6 y + 21 z = 114 \\0.6 x + 0.4 y + 1.4 z = 7.6\end{array}3x+2y+7z=389x+6y+21z=1140.6x+0.4y+1.4z=7.6​

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - −4x−6y+5z=138x+4y−7z=−74x−2y−2z=−15\begin{array} { l } - 4 x - 6 y + 5 z = 13 \\8 x + 4 y - 7 z = - 7 \\4 x - 2 y - 2 z = - 15\end{array}−4x−6y+5z=138x+4y−7z=−74x−2y−2z=−15​

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - −2x−y+3z=13x−3y−3z=−4\begin{array} { l } - 2 x - y + 3 z = 13 \\x - 3 y - 3 z = - 4\end{array}−2x−y+3z=13x−3y−3z=−4​

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -True or false? A system of linear equations in three variables may have infinitely many solutions.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If the ________of a matrix is p×qp \times qp×q , then p represents the number of________ and q represents the number of .

Determine the solution set for the system represented by the augmented matrix. - [10−47018−70000]\left[ \begin{array} { r r r | r } 1 & 0 & - 4 & 7 \\ 0 & 1 & 8 & - 7 \\ 0 & 0 & 0 & 0 \end{array} \right]​100​010​−480​7−70​​

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - −3x−7y+7z=71−2x+7y−3z=−22−3x−9y+9z=87\begin{array} { l } - 3 x - 7 y + 7 z = 71 \\- 2 x + 7 y - 3 z = - 22 \\- 3 x - 9 y + 9 z = 87\end{array}−3x−7y+7z=71−2x+7y−3z=−22−3x−9y+9z=87​

The solution set to a system of dependent equations is given. Write the specific solution corresponding to the given value of z. - $\{ ( 4 z - 7 , z - 4 , z ) \mid z$ is any real number $\}$ $z = - 5$

Determine the solution set for the system represented by the augmented matrix. - $\left[ \begin{array} { r r r | r } 1 & 0 & - 7 & 4 \\ 0 & 1 & 6 & 1 \\ 0 & 0 & 0 & 1 \end{array} \right]$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } - 6 x + 15 y = 15 \\2 x - 5 y = - 5\end{array}$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } 8 x + 10 y = - 17 \\4 x + 5 y = - 9\end{array}$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } 3 x + 2 y + 7 z = 38 \\9 x + 6 y + 21 z = 114 \\0.6 x + 0.4 y + 1.4 z = 7.6\end{array}$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } - 4 x - 6 y + 5 z = 13 \\8 x + 4 y - 7 z = - 7 \\4 x - 2 y - 2 z = - 15\end{array}$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } - 2 x - y + 3 z = 13 \\x - 3 y - 3 z = - 4\end{array}$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If the of a matrix is $p \times q$ , then p represents the number of and q represents the number of .

Determine the solution set for the system represented by the augmented matrix. - $\left[ \begin{array} { r r r | r } 1 & 0 & - 4 & 7 \\ 0 & 1 & 8 & - 7 \\ 0 & 0 & 0 & 0 \end{array} \right]$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } - 3 x - 7 y + 7 z = 71 \\- 2 x + 7 y - 3 z = - 22 \\- 3 x - 9 y + 9 z = 87\end{array}$