In order to solve the following system of equations by using technology to reduce an augmented matrix to row echelon form, how many rows and how many columns are needed?

Question

⎧
⎪
⎨
⎪
⎩
x
+
y
=
4
2
x
−
3
z
=
14
2
y
+
z
=
2


(1 point)
Responses

4 rows, 3 columns
4 rows, 3 columns

3 rows, 3 columns
3 rows, 3 columns

3 rows, 4 columns
3 rows, 4 columns

A matrix cannot be created, as variables are missing.
A matrix cannot be created, as variables are missing.

Answer 1

To represent the given system of equations as an augmented matrix, we first identify the number of variables and the number of equations.

The system of equations is:

1. \( x + y = 4 \)
2. \( 2x - 3z = 14 \)
3. \( 2y + z = 2 \)

Step 1: Count Variables

The variables present in the equations are:

• \( x \)
• \( y \)
• \( z \)

This gives us 3 variables.

Step 2: Count Equations

There are 3 equations in total.

Step 3: Construct the Augmented Matrix

The augmented matrix will consist of:

• One row for each equation (3 rows)
• One column for each variable (3 columns)
• An additional column for the constants on the right side of the equations.

Augmented Matrix Structure

1. The first row corresponds to the first equation \( x + y = 4 \), which can be represented as \( [1, 1, 0 | 4] \).
2. The second row corresponds to the second equation \( 2x - 3z = 14 \), represented as \( [2, 0, -3 | 14] \).
3. The third row corresponds to the third equation \( 2y + z = 2 \), represented as \( [0, 2, 1 | 2] \).

Final Augmented Matrix

The augmented matrix will look like this:

\[ \begin{bmatrix} 1 & 1 & 0 & | & 4 \ 2 & 0 & -3 & | & 14 \ 0 & 2 & 1 & | & 2 \end{bmatrix} \]

Conclusion

The augmented matrix has:

• 3 rows (one for each equation)
• 4 columns (one for each variable and one for the constants)

Therefore, the correct answer is: 3 rows, 4 columns.