Question 1

Which of the following best describes a system of equations that has exactly one solution?

Accepted Answer

Consistent and independent

Question 2

Given the augmented matrix below, which condition for reduced row echelon form is violated?

$$
\begin{bmatrix}
1 & 2 & 0 \
1 & 0 & 3
\end{bmatrix}
$$

Accepted Answer

Condition 3: All other elements in the column containing the leftmost 1 of a given row are zeros.

Question 3

Which row operation would you use to transform the following matrix into reduced row echelon form?

$$
\begin{bmatrix}
0 & 1 & 2 \
1 & 0 & 3
\end{bmatrix}
$$

Accepted Answer

Interchange row 1 and row 2

Question 4

A system of equations is represented by the following augmented matrix:

$$
\begin{bmatrix}
1 & 0 & 2 & | & 3 \
0 & 1 & -1 & | & 4 \
0 & 0 & 0 & | & 5
\end{bmatrix}
$$

What can you conclude about the system?

Accepted Answer

The system is inconsistent and has no solution.

Question 5

Which of the following is NOT a valid row operation in Gauss-Jordan elimination?

Accepted Answer

Multiplying one row by another row

Question 6

A company wants to buy 25 trucks with a total capacity of 28,000 cubic feet. The trucks available have capacities of 350, 700, and 1,400 cubic feet. If $$x$$, $$y$$, and $$z$$ represent the number of each type of truck, which equation models the total capacity constraint?

Accepted Answer

$$350x + 700y + 1400z = 28000$$

Question 7

Suppose you have the following reduced matrix:

$$
\begin{bmatrix}
1 & 0 & 2 & | & 3 \
0 & 1 & -1 & | & 4 \
0 & 0 & 0 & | & 0
\end{bmatrix}
$$

How many solutions does the corresponding system have?

Accepted Answer

Infinitely many solutions

Question 8

Which step comes first in the three-step model for solving real-world problems with mathematics?

Accepted Answer

Construct a mathematical model

Question 9

Given the system:

$$
\begin{cases}
x + y + z = 6 \
x - y + z = 2 \
2x + y - z = 3
\end{cases}
$$

What is the first step in writing the augmented matrix for this system?

Accepted Answer

Write the coefficients and constants in a matrix as rows

Question 10

If a reduced matrix has fewer leftmost 1's than variables and no contradictions, what does this imply about the system?

Accepted Answer

The system is dependent and has infinitely many solutions.

Question 11

A real-world problem is modeled by the system:

$$
\begin{cases}
x + y + z = 25 \
350x + 700y + 1400z = 28000
\end{cases}

If$$ $$z = 10$$, what is the value of $$x + y$$?

Accepted Answer

$$x + y = 15$$

Question 12

Which of the following is a necessary condition for a matrix to be in reduced row echelon form?

Accepted Answer

Each leading 1 is the only nonzero entry in its column.

Question 13

You are solving a system using Gauss-Jordan elimination and obtain the row:

$$
\begin{bmatrix} 0 & 0 & 0 & | & 7 \end{bmatrix}
$$

What does this indicate about the system?

Accepted Answer

The system is inconsistent and has no solution.

Question 14

Which of the following best describes the purpose of Gauss-Jordan elimination?

Accepted Answer

To transform a system of equations into reduced row echelon form to easily identify solutions

Question 15

A system of equations is modeled by the following augmented matrix:

$$
\begin{bmatrix}
1 & 0 & 2 & | & 3 \
0 & 1 & -1 & | & 4 \
0 & 0 & 0 & | & 0
\end{bmatrix}

If$$ $$z = t$$, express $$x$$ and $$y in $$terms of $$t.$$

Accepted Answer

$$x = 3 - 2t,\ y = 4 + t$$

Systems of Linear Equations and Gauss-Jordan Elimination: Study Notes

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