Math Problem Statement
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
Solution
To guarantee that the given linear system is consistent, we need to determine the value of that ensures the system has at least one solution.
The system of equations is:
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.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Gaussian Elimination
Consistency in Linear Systems
Formulas
Gaussian elimination
Row-reduction techniques
Theorems
Existence and Uniqueness Theorem for Linear Systems
Rank-Nullity Theorem
Suitable Grade Level
Grades 10-12
Related Recommendation
Solve and Test Consistency of System of Linear Equations Using Gaussian Elimination
Gaussian Elimination: Determine Consistency and Solution of a 3x3 System of Linear Equations
Determine b1, b2, b3 for a Consistent Linear System
Ax = b: Solving Systems of Linear Equations and Describing Solution Sets
Solving a System of Linear Equations using Gaussian Elimination