Question

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?

⎧
⎪
⎨
⎪
⎩
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**.