Q: Which of the following systems of equations is classified as inconsistent?

$$\begin{cases} y = 2x + 3 \\ y = 2x - 4 \end{cases}$$

Q: Which of the following augmented matrices represents the system $$\begin{cases} 2x + 3y = 5 \\ 4x - y = 7 \end{cases}$$?

$$\begin{bmatrix} 2 & 3 & 5 \\ 4 & -1 & 7 \end{bmatrix}$$

Q: Which of the following systems is consistent and independent?

$$\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}$$

Q: Which of the following is the correct augmented matrix for the system $$\begin{cases} 3x + 2y - z = 4 \\ x - y + 2z = 5 \\ 2x + y + z = 6 \end{cases}$$?

$$\begin{bmatrix} 3 & 2 & -1 & 4 \\ 1 & -1 & 2 & 5 \\ 2 & 1 & 1 & 6 \end{bmatrix}$$

Q: Which of the following systems of inequalities would have a solution region that is a bounded area (polygon) on the graph?

$$\begin{cases} y \u003e x + 1 \\ y 0 \end{cases}$$

Question 1

Which of the following systems of equations is classified as inconsistent?

Accepted Answer

$$\begin{cases} y = 2x + 3 \ y = 2x - 4 \end{cases}$$

Question 2

Given the system $$\begin{cases} 4x + 2y = 7 \ x + 5y = 4 \end{cases}$$, what is the value of $$y$$ using the substitution method?

Accepted Answer

$$y = 0.6$$

Question 3

Which matrix operation corresponds to swapping two rows in a matrix?

Accepted Answer

$$R_i \leftrightarrow R_j$$

Question 4

What is the determinant of the matrix $$\begin{bmatrix} 3 & 5 \ 2 & 4 \end{bmatrix}$$?

Accepted Answer

$$2$$

Question 5

A system of equations is called dependent if:

Accepted Answer

It has infinitely many solutions.

Question 6

Given the system $$\begin{cases} 2x + 3y = 1 \ 3x - y = 4 \end{cases}$$, what is the solution for $$x$$ using the elimination method?

Accepted Answer

$$x = 1.4$$

Question 7

Which of the following augmented matrices represents the system $$\begin{cases} 2x + 3y = 5 \ 4x - y = 7 \end{cases}$$?

Accepted Answer

$$\begin{bmatrix} 2 & 3 & 5 \ 4 & -1 & 7 \end{bmatrix}$$

Question 8

If you multiply the first row of a matrix by $$-2$$, which row operation notation is correct?

Accepted Answer

$$R_1$$ 	o -2R_1$$

Question 9

Using Cramer's Rule, what is the value of $$x$$ for the system $$\begin{cases} x + 2y = 5 \ 3x - y = 4 \end{cases}$$?

Accepted Answer

$$x = 3$$

Question 10

Which of the following best describes the solution region for the system of inequalities $$y \u003e 2x - 1$$ and $$y \u003c -x + 4$$?

Accepted Answer

The region between the two lines.

Question 11

What is the determinant of the $$3 	imes 3$$ matrix $$\begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 4 \ 5 & 6 & 0 \end{bmatrix}$$?

Accepted Answer

$$1$$

Question 12

Which of the following systems is consistent and independent?

Accepted Answer

$$\begin{cases} y = 2x + 1 \ y = -x + 4 \end{cases}$$

Question 13

Which row operation would you use to eliminate $$y in $$the system $$\begin{cases} 2x + 3y = 7 \ 4x + 6y = 14 \end{cases}$$?

Accepted Answer

Multiply the first equation by $$2$$ and subtract from the second.

Question 14

Given the system $$\begin{cases} x + y = 3 \ x - y = 1 \end{cases}$$, what is the solution for $$(x, y)$$?

Accepted Answer

$$(2, 1)$$

Question 15

Which of the following is the correct augmented matrix for the system $$\begin{cases} 3x + 2y - z = 4 \ x - y + 2z = 5 \ 2x + y + z = 6 \end{cases}$$?

Accepted Answer

$$\begin{bmatrix} 3 & 2 & -1 & 4 \ 1 & -1 & 2 & 5 \ 2 & 1 & 1 & 6 \end{bmatrix}$$

Question 16

Which of the following is NOT a valid row operation for matrices?

Accepted Answer

Multiplying a row by zero.

Question 17

If the determinant of a $$2 	imes 2$$ matrix is zero, what does this imply about the corresponding system of equations?

Accepted Answer

The system has no solution or infinitely many solutions.

Question 18

Which of the following systems of inequalities would have a solution region that is a bounded area (polygon) on the graph?

Accepted Answer

$$\begin{cases} y \u003e x + 1 \ y  0 \end{cases}$$

Question 19

Which of the following is the correct solution to the system $$\begin{cases} y = 3x - 2 \ 2x + y = 4 \end{cases}$$?

Accepted Answer

$$(2, 4)$$

Systems of Equations and Matrices – Precalculus Study Guide

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