Math Problem Statement
Solve the system of equation by Gauss Jordan method: 2x + 3y + 4z = 20; x + 2y + z = 8; 4x + y + 2z = 12.
Solution
To solve the given system of equations using the Gauss-Jordan method, we start by expressing the system as an augmented matrix:
System of equations:
Augmented matrix:
2 & 3 & 4 & | & 20 \\ 1 & 2 & 1 & | & 8 \\ 4 & 1 & 2 & | & 12 \end{bmatrix}$$ We will use row operations to transform this matrix into reduced row-echelon form. --- ### Step 1: Normalize the first row. Divide the first row by 2 to make the pivot element in the first row, first column equal to 1: $$\begin{bmatrix} 1 & 1.5 & 2 & | & 10 \\ 1 & 2 & 1 & | & 8 \\ 4 & 1 & 2 & | & 12 \end{bmatrix}$$ --- ### Step 2: Eliminate the first column of rows 2 and 3. - For row 2: $$R_2 \to R_2 - R_1$$ - For row 3: $$R_3 \to R_3 - 4R_1$$ Updated matrix: $$\begin{bmatrix} 1 & 1.5 & 2 & | & 10 \\ 0 & 0.5 & -1 & | & -2 \\ 0 & -5 & -6 & | & -28 \end{bmatrix}$$ --- ### Step 3: Normalize the second row. Divide the second row by 0.5 to make the pivot element in the second row, second column equal to 1: $$\begin{bmatrix} 1 & 1.5 & 2 & | & 10 \\ 0 & 1 & -2 & | & -4 \\ 0 & -5 & -6 & | & -28 \end{bmatrix}$$ --- ### Step 4: Eliminate the second column of rows 1 and 3. - For row 1: $$R_1 \to R_1 - 1.5R_2$$ - For row 3: $$R_3 \to R_3 + 5R_2$$ Updated matrix: $$\begin{bmatrix} 1 & 0 & 5 & | & 16 \\ 0 & 1 & -2 & | & -4 \\ 0 & 0 & -16 & | & -48 \end{bmatrix}$$ --- ### Step 5: Normalize the third row. Divide the third row by $$-16$$ to make the pivot element in the third row, third column equal to 1: $$\begin{bmatrix} 1 & 0 & 5 & | & 16 \\ 0 & 1 & -2 & | & -4 \\ 0 & 0 & 1 & | & 3 \end{bmatrix}$$ --- ### Step 6: Eliminate the third column of rows 1 and 2. - For row 1: $$R_1 \to R_1 - 5R_3$$ - For row 2: $$R_2 \to R_2 + 2R_3$$ Final matrix: $$\begin{bmatrix} 1 & 0 & 0 & | & 1 \\ 0 & 1 & 0 & | & 2 \\ 0 & 0 & 1 & | & 3 \end{bmatrix}$$ --- ### Solution: The solution to the system of equations is: $$x = 1, \quad y = 2, \quad z = 3$$ --- Do you want a detailed explanation of any specific step or further clarifications? --- ### Related Questions: 1. How does the Gauss-Jordan method differ from Gaussian elimination? 2. What are common pitfalls in using the Gauss-Jordan method? 3. How do we interpret cases where the system has no solution or infinitely many solutions? 4. Can you explain why we normalize rows to create pivot elements? 5. How does scaling rows affect the solution of the system? --- ### Tip: Always double-check your arithmetic during row operations to avoid errors that can propagate through the solution process.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Gauss-Jordan Elimination
Matrix Operations
Formulas
Gauss-Jordan Elimination Method
Theorems
Gauss-Jordan Elimination Theorem
Suitable Grade Level
Grades 10-12
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