Test for consistency of the following system of linear equations and solve
a. 3𝑥 + 3𝑦 + 2𝑧 = 1, 𝑥 + 2𝑦 = 4, 10𝑦 + 3𝑧 = −2, 2𝑥 − 3𝑦 − 𝑧 = 5

To test for the consistency of the system of linear equations and solve them, we will use the following steps:

### Step 1: Write the system of equations
Given:
1. \( 3x + 3y + 2z = 1 \)  \(\quad\)  (Equation 1)
2. \( x + 2y = 4 \) \(\quad\) \(\quad\) \(\quad\quad\) (Equation 2)
3. \( 10y + 3z = -2 \)  \(\quad\quad\) (Equation 3)
4. \( 2x - 3y - z = 5 \) \(\quad\quad\quad\) (Equation 4)

### Step 2: Express the system as augmented matrix
We will represent the system as an augmented matrix:

\[
\begin{pmatrix}
3 & 3 & 2 & | & 1 \\
1 & 2 & 0 & | & 4 \\
0 & 10 & 3 & | & -2 \\
2 & -3 & -1 & | & 5
\end{pmatrix}
\]

### Step 3: Perform Gaussian elimination
We will now perform row operations to reduce the matrix to row echelon form and check for consistency.

I'll perform the calculations to solve this system.It appears that the system of equations cannot be directly solved using the standard approach, likely due to the system being either inconsistent or having no unique solution. Let's proceed by further analyzing the augmented matrix to check for consistency manually and continue the row reduction process.The reduced row echelon form (RREF) of the augmented matrix is:

\[
\begin{pmatrix}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & -4 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

This result shows that the system is **consistent** since there are no contradictory rows (such as \(0 = 1\)) in the matrix.

### Solution:
From the RREF matrix, we can directly read the solutions:
- \( x = 2 \)
- \( y = 1 \)
- \( z = -4 \)

Thus, the solution to the system of equations is:
\[
x = 2, \quad y = 1, \quad z = -4
\]

Would you like a detailed explanation of the row operations, or do you have any questions?

Here are five questions to further explore:
1. What is the significance of the row echelon form in solving linear systems?
2. How do you interpret a system of linear equations with a row of zeros in the augmented matrix?
3. What are the conditions for a system to be inconsistent?
4. How can Gaussian elimination be used to check for unique or infinite solutions?
5. What are the alternative methods to solve a system of linear equations?

**Tip:** When reducing matrices, avoid arithmetic mistakes by carefully choosing elementary row operations.

Test for consistency of the following system of linear equations and solve: a. 3x + 3y + 2z = 1, x + 2y = 4, 10y + 3z = −2, 2x − 3y − z = 5

Learn how to test for consistency and solve a system of linear equations using Gaussian elimination. This problem covers core linear algebra concepts, including matrix row reduction and checking for consistency through the RREF method. Suitable for students in Grades 10-12.

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