Q: Given the system of equations: $$2x + 5y + z = 1$$ $$x + 2y - z = -3$$ $$-x + 4y - 2z = 3$$ Which of the following is the correct reduced row echelon form (RREF) of the augmented matrix for this system?

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

Q: Given the system: $$x + 3y + 4z = 11$$ $$3x + 15y + 10z = 15$$ $$2x + 11y + 5z = 3$$ What is the RREF of the augmented matrix for this system?

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

Question 1

Given the system of equations:

$$2x + 5y + z = 1$$

$$x + 2y - z = -3$$

$$-x + 4y - 2z = 3$$

Which of the following is the correct reduced row echelon form (RREF) of the augmented matrix for this system?

Accepted Answer

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

Question 2

A chemist is mixing three solutions with concentrations $$x$$, $$y$$, and $$z to $$obtain a final mixture. The system of equations representing the mixture is:

$$2x + 7y + z = 13$$

$$x + 3y - z = 9$$

$$-x + y - z = -1$$

What is the value of $$y in $$the solution to this system?

Accepted Answer

$$y = 3$$

Question 3

Given the system:

$$2x + 7y + 6z = 4$$

$$x + 4y + 5z = 3$$

$$-x - 2y + 3z = -1$$

After performing row reduction, which of the following statements is true about the solution set?

Accepted Answer

The system has a unique solution.

Question 4

A medical researcher is analyzing the effect of three drugs, $$x$$, $$y$$, and $$z$$, on a disease. The system is:

$$3x + y + 2z = 16$$

$$2x + 2y + 3z = 16$$

$$2x - y + z = 11$$

What is the value of $$z in $$the solution to this system?

Accepted Answer

$$z = 3$$

Question 5

Given the system:

$$x + 3y + 4z = 11$$

$$3x + 15y + 10z = 15$$

$$2x + 11y + 5z = 3$$

What is the RREF of the augmented matrix for this system?

Accepted Answer

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

Question 6

Which row operation would you use to eliminate the $$x-$$term from the second equation in the system:

$$2x + 5y + z = 1$$

$$x + 2y - z = -3$$

$$-x + 4y - 2z = 3$$

Accepted Answer

Replace $$R_2$$ with $$R_2$$ - $$R_1$$

Question 7

After obtaining the RREF of a $$3 	imes 4$$ augmented matrix, you find a row of the form $$[0\ 0\ 0\ |\ 1]. $$What does this indicate about the system of equations?

Accepted Answer

The system has no solution.

Question 8

Suppose you are solving a system of three equations in three variables and, after row reduction, you obtain the following RREF:

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

What does this tell you about the solution set?

Accepted Answer

There are infinitely many solutions.

Question 9

A system of equations is represented by the augmented matrix:

$$\begin{bmatrix}2 & 7 & 1 & 13 \ 1 & 3 & -1 & 9 \ -1 & 1 & -1 & -1\end{bmatrix}$$

What is the first step to begin row reduction to RREF?

Accepted Answer

Swap $$R_1$$ and $$R_2 to $$get a leading 1 in the first row.

Question 10

In the context of balancing chemical equations, a system of equations is set up for the coefficients of the reactants and products. If the RREF of the augmented matrix has a free variable, what does this imply about the chemical equation?

Accepted Answer

There are infinitely many ways to balance the equation, depending on the value of the free variable.

Question 11

A system of equations is given by:

$$2x + 7y + 6z = 4$$

$$x + 4y + 5z = 3$$

$$-x - 2y + 3z = -1$$

After row reducing, you find $$z = 1. $$What are the values of $$x$$ and $$y$$?

Accepted Answer

$$x = -1,\ y = 1$$

Question 12

Which of the following is NOT a valid elementary row operation when performing row reduction?

Accepted Answer

Multiplying one row by another row.

Question 13

A system of equations is represented by the augmented matrix:

$$\begin{bmatrix}1 & 3 & 4 & 11 \ 3 & 15 & 10 & 15 \ 2 & 11 & 5 & 3\end{bmatrix}$$

After row reduction, the solution is $$x = 1$$, $$y = 1$$, $$z = 2. $$Which of the following is a correct interpretation?

Accepted Answer

The system has a unique solution.

Question 14

If the RREF of an augmented matrix for a system of three equations in three variables is:

$$\begin{bmatrix}1 & 0 & 0 & a \ 0 & 1 & 0 & b \ 0 & 0 & 1 & c\end{bmatrix}$$

What does this form indicate about the system?

Accepted Answer

The system has a unique solution: $$x = a$$, $$y = b$$, $$z = c.$$

Question 15

A business is solving for the optimal mix of three products, $$x$$, $$y$$, and $$z$$, using the system:

$$3x + y + 2z = 16$$

$$2x + 2y + 3z = 16$$

$$2x - y + z = 11

If $$the RREF of the augmented matrix is:

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

What is the optimal mix?

Accepted Answer

$$x = 2,\ y = 3,\ z = 3$$

Question 16

When row reducing the augmented matrix for a system, why is it important to obtain leading 1's in each row?

Accepted Answer

To ensure each variable can be solved for directly and to reach RREF.

Question 17

Suppose you have the following system:

$$2x + 7y + z = 13$$

$$x + 3y - z = 9$$

$$-x + y - z = -1$$

After row reduction, you find $$x = 2$$, $$y = 3$$, $$z = 1. $$Which equation would you use to check your solution?

Accepted Answer

Any of the original equations can be used to check the solution.

Question 18

If, after row reduction, the RREF of a $$3 	imes 4$$ augmented matrix is:

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

What does this indicate about the solution set?

Accepted Answer

There are infinitely many solutions, with one free variable.

Question 19

A student row reduces a $$3 	imes 4$$ augmented matrix and gets:

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

What is the solution to the original system?

Accepted Answer

$$x = 2,\ y = -1,\ z = 1$$

Step-by-Step Guidance: Row Reduction to RREF for Systems of Linear Equations LO9

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