Math Problem Statement
Given the linear system x2− 4x3 =8 2x1− 3x2 + 2x3 =1 5x1− 8x2 + 7x3 =1 a Rewrite the system in matrix notation. b Using elementary row operations only, reduce the system to an upper triangular matrix. Hint: You may need to swap rows to achieve this. c Using the upper triangular matrix, solve for x1, x2, and x3. Note: for an inconsistent system, you will not be able to solve the system. d Is the system consistent?
Solution
Let’s tackle the problem step by step.
a) Rewrite the system in matrix notation
The system of equations:
can be written in matrix form as:
0 & 1 & -4 \\ 2 & -3 & 2 \\ 5 & -8 & 7 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 8 \\ 1 \\ 1 \end{bmatrix}$$ --- ### b) Reduce the system to an upper triangular matrix We will perform row operations to reduce the matrix: $$\left[ \begin{array}{ccc|c} 0 & 1 & -4 & 8 \\ 2 & -3 & 2 & 1 \\ 5 & -8 & 7 & 1 \end{array} \right]$$ #### Step 1: Swap $$R_1$$ and $$R_2$$ (so the pivot is non-zero in the first row): $$\left[ \begin{array}{ccc|c} 2 & -3 & 2 & 1 \\ 0 & 1 & -4 & 8 \\ 5 & -8 & 7 & 1 \end{array} \right]$$ #### Step 2: Eliminate $$x_1$$ from $$R_3$$ using $$R_1$$: $$R_3 \to R_3 - \frac{5}{2}R_1$$ New matrix: $$\left[ \begin{array}{ccc|c} 2 & -3 & 2 & 1 \\ 0 & 1 & -4 & 8 \\ 0 & -\frac{1}{2} & 2 & -\frac{3}{2} \end{array} \right]$$ #### Step 3: Eliminate $$x_1$$ from $$R_2$$ (already 0, so no change). #### Step 4: Swap $$R_2$$ and $$R_3$$ (to simplify further operations): $$\left[ \begin{array}{ccc|c} 2 & -3 & 2 & 1 \\ 0 & -\frac{1}{2} & 2 & -\frac{3}{2} \\ 0 & 1 & -4 & 8 \end{array} \right]$$ #### Step 5: Scale $$R_2$$ to make the pivot 1: $$R_2 \to -2R_2$$ New matrix: $$\left[ \begin{array}{ccc|c} 2 & -3 & 2 & 1 \\ 0 & 1 & -4 & 3 \\ 0 & 1 & -4 & 8 \end{array} \right]$$ #### Step 6: Eliminate $$x_2$$ from $$R_3$$: $$R_3 \to R_3 - R_2$$ New matrix: $$\left[ \begin{array}{ccc|c} 2 & -3 & 2 & 1 \\ 0 & 1 & -4 & 3 \\ 0 & 0 & 0 & 5 \end{array} \right]$$ --- ### c) Solve for $$x_1, x_2, x_3$$ From the final matrix, we see: $$\left[ \begin{array}{ccc|c} 2 & -3 & 2 & 1 \\ 0 & 1 & -4 & 3 \\ 0 & 0 & 0 & 5 \end{array} \right]$$ The last row indicates: $$0 = 5$$ This is a contradiction, so the system is **inconsistent**. --- ### d) Is the system consistent? No, the system is **inconsistent** because the last row produces a contradiction. --- ### Questions for Further Exploration: 1. What are some real-world examples where inconsistent systems arise? 2. How can a matrix be augmented to ensure consistency in a linear system? 3. Can the rank of a matrix help determine inconsistency without solving? 4. How does Gaussian elimination differ from Gauss-Jordan elimination? 5. Are there alternative methods to identify inconsistency (e.g., determinant, rank)? **Tip:** Always inspect the augmented column for contradictions when performing row reductions to detect inconsistency quickly.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Systems
Matrix Notation
Row Operations
Gaussian Elimination
Matrix Algebra
Formulas
Ax = b
Elementary Row Operations
Upper Triangular Matrix
Gaussian Elimination
Theorems
Row Echelon Form
Existence and Uniqueness Theorem
Suitable Grade Level
Grades 10-12
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