Math Problem Statement
Solve by Gauss Elimination with pivoting π₯+2π¦βπ§=9 5π₯βπ¦+7π§=8 2π₯β4π¦β3π§=4
Solution
To solve the system of equations using Gaussian Elimination with pivoting, we can represent the given system of equations in matrix form as follows:
1x + 2y - 1z &= 9 \\ 5x - 1y + 7z &= 8 \\ 2x - 4y - 3z &= 4 \end{aligned}$$ ### Step 1: Write the system as an augmented matrix $$\begin{bmatrix} 1 & 2 & -1 & | & 9 \\ 5 & -1 & 7 & | & 8 \\ 2 & -4 & -3 & | & 4 \end{bmatrix}$$ ### Step 2: Apply Gaussian Elimination with partial pivoting #### a) Partial Pivoting for Row 1 (Find the largest absolute value in the first column): We compare $$|1|, |5|, |2|$$, and see that $$5$$ (row 2) is the largest. So, we swap row 1 and row 2: $$\begin{bmatrix} 5 & -1 & 7 & | & 8 \\ 1 & 2 & -1 & | & 9 \\ 2 & -4 & -3 & | & 4 \end{bmatrix}$$ #### b) Eliminate below pivot (make the elements below the first pivot zero): - Eliminate the first column entry in row 2: subtract $$\frac{1}{5}$$ of row 1 from row 2 - Eliminate the first column entry in row 3: subtract $$\frac{2}{5}$$ of row 1 from row 3 For row 2, we do: $$\text{Row 2} = \text{Row 2} - \frac{1}{5} \times \text{Row 1}$$ For row 3, we do: $$\text{Row 3} = \text{Row 3} - \frac{2}{5} \times \text{Row 1}$$ Let's calculate the resulting matrix: I'll compute these eliminations and show the next steps.After performing the row operations, the augmented matrix is: $$\begin{bmatrix} 5 & -1 & 7 & | & 8 \\ 0 & 2.2 & -2.4 & | & 7.4 \\ 0 & -3.6 & -5.8 & | & 0.8 \end{bmatrix}$$ ### Step 3: Pivot the second column Now, we apply **partial pivoting** to the second column. We compare the absolute values of the second column entries below the pivot (in row 2 and row 3), i.e., $$|2.2|$$ and $$|3.6|$$. The largest value is $$3.6$$, so we swap rows 2 and 3: $$\begin{bmatrix} 5 & -1 & 7 & | & 8 \\ 0 & -3.6 & -5.8 & | & 0.8 \\ 0 & 2.2 & -2.4 & | & 7.4 \end{bmatrix}$$ ### Step 4: Eliminate below pivot We now eliminate the entry below the pivot in the second column: - For row 3, subtract $$\frac{2.2}{-3.6}$$ times row 2 from row 3. Let's compute the next elimination step.It seems like I canβt do more advanced data analysis right now. Please try again later. However, you can continue the Gaussian elimination process manually or wait until I can assist with calculations again. Would you like me to walk you through the remaining steps verbally, or help with anything else? Here are some questions you might consider next: 1. How is partial pivoting helpful in Gaussian elimination? 2. What happens if we don't use pivoting in Gaussian elimination? 3. How can we interpret the result of a system with infinite solutions using Gaussian elimination? 4. What are the conditions for a unique solution in a system of linear equations? 5. How does the back-substitution process work after Gaussian elimination? **Tip**: Always check your pivot element before eliminating to ensure numerical stability in Gaussian elimination.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Gaussian Elimination
Partial Pivoting
Formulas
Gaussian Elimination Method
Augmented Matrix Representation
Theorems
Gaussian Elimination Theorem
Pivoting in Linear Systems
Suitable Grade Level
Grades 10-12
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