
Determine the value of b that would guarantee that the below linear system is consistent.

x1 − 4x2 − 6x3	 = 	−4
5x1 − 6x2 − 4x3	 = 	7
−11x1 + 2x2 − 12x3	 = 	b

To guarantee that the given linear system is consistent, we need to determine the value of \( b \) that ensures the system has at least one solution.

The system of equations is:

\[
\begin{aligned}
1. \quad x_1 - 4x_2 - 6x_3 &= -4 \\
2. \quad 5x_1 - 6x_2 - 4x_3 &= 7 \\
3. \quad -11x_1 + 2x_2 - 12x_3 &= b
\end{aligned}
\]

### Step 1: Write the system in matrix form
We can express this system in matrix form as:

\[
\begin{bmatrix}
1 & -4 & -6 \\
5 & -6 & -4 \\
-11 & 2 & -12
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}
=
\begin{bmatrix}
-4 \\
7 \\
b
\end{bmatrix}
\]

### Step 2: Analyze the augmented matrix

The augmented matrix is:

\[
\begin{bmatrix}
1 & -4 & -6 & -4 \\
5 & -6 & -4 & 7 \\
-11 & 2 & -12 & b
\end{bmatrix}
\]

### Step 3: Perform row reduction to simplify

We will reduce the coefficient matrix to echelon form to find conditions for \( b \) to guarantee consistency. Let's apply Gaussian elimination.

#### Row 1: \( R_1 = [1, -4, -6, -4] \)

#### Row 2: Eliminate \( x_1 \)-term in \( R_2 \) using \( R_1 \):

\[
R_2 \rightarrow R_2 - 5 \times R_1 = [5, -6, -4, 7] - 5 \times [1, -4, -6, -4] = [0, 14, 26, 27]
\]

#### Row 3: Eliminate \( x_1 \)-term in \( R_3 \) using \( R_1 \):

\[
R_3 \rightarrow R_3 + 11 \times R_1 = [-11, 2, -12, b] + 11 \times [1, -4, -6, -4] = [0, -42, -78, b - 44]
\]

At this point, the matrix is:

\[
\begin{bmatrix}
1 & -4 & -6 & -4 \\
0 & 14 & 26 & 27 \\
0 & -42 & -78 & b - 44
\end{bmatrix}
\]

#### Row 3: Eliminate \( x_2 \)-term in \( R_3 \) using \( R_2 \):

\[
R_3 \rightarrow R_3 + 3 \times R_2 = [0, -42, -78, b - 44] + 3 \times [0, 14, 26, 27] = [0, 0, 0, b + 37]
\]

### Step 4: Conclusion from row-reduction

The third row of the augmented matrix is:

\[
[0, 0, 0, b + 37]
\]

For the system to be consistent, the last element must be zero:

\[
b + 37 = 0
\]

Thus, solving for \( b \):

\[
b = -37
\]

### Final Answer:

The value of \( b \) that guarantees the system is consistent is \( \boxed{-37} \).

---

Would you like more details on Gaussian elimination or other questions related to this system? Here are some related questions:

1. What is Gaussian elimination and how does it help solve linear systems?
2. Can the system have a unique solution for \( b = -37 \)?
3. How do you determine if a system is inconsistent?
4. What is the geometric interpretation of consistent linear systems?
5. What is the rank of a matrix and how does it relate to the solution of a system?

**Tip**: Consistency in linear systems is often determined by checking for contradictions in the augmented matrix after row reduction.

Determine the value of b that would guarantee that the below linear system is consistent.

x1 − 4x2 − 6x3 = −4
5x1 − 6x2 − 4x3 = 7
−11x1 + 2x2 − 12x3 = b

Explore how to find the value of b that guarantees the consistency of a linear system of equations using Gaussian elimination. This problem covers linear algebra concepts like systems of linear equations, row-reduction techniques, and matrix operations. Suitable for students in Grades 10-12.

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