Math Problem Statement
Solve the following system of equations using Gauss Jorden Method: (i). x + 2y + 6z = 44, 3x + 4y + z = 52, 6x − y − z = 38
Solution
To solve the given system of linear equations using the Gauss-Jordan Method, we first represent the system in augmented matrix form, then perform row operations to reduce the matrix to reduced row-echelon form.
Given Equations:
Step 1: Write the augmented matrix
1 & 2 & 6 & 44 \\ 3 & 4 & 1 & 52 \\ 6 & -1 & -1 & 38 \end{bmatrix}$$ ### Step 2: Perform row operations We will use the following steps to reduce the matrix. --- #### **Step 2.1:** Make the pivot element in row 1, column 1, equal to 1 (already 1). Use row 1 to make other entries in column 1 zero. - Subtract $$3R_1$$ from $$R_2$$: $$R_2 \rightarrow R_2 - 3R_1$$ - Subtract $$6R_1$$ from $$R_3$$: $$R_3 \rightarrow R_3 - 6R_1$$ New matrix: $$\begin{bmatrix} 1 & 2 & 6 & 44 \\ 0 & -2 & -17 & -80 \\ 0 & -13 & -37 & -226 \end{bmatrix}$$ --- #### **Step 2.2:** Make the pivot element in row 2, column 2, equal to 1. - Divide $$R_2$$ by $$-2$$: $$R_2 \rightarrow \frac{R_2}{-2}$$ New matrix: $$\begin{bmatrix} 1 & 2 & 6 & 44 \\ 0 & 1 & \frac{17}{2} & 40 \\ 0 & -13 & -37 & -226 \end{bmatrix}$$ --- #### **Step 2.3:** Use $$R_2$$ to eliminate other entries in column 2. - Subtract $$2R_2$$ from $$R_1$$: $$R_1 \rightarrow R_1 - 2R_2$$ - Add $$13R_2$$ to $$R_3$$: $$R_3 \rightarrow R_3 + 13R_2$$ New matrix: $$\begin{bmatrix} 1 & 0 & -11 & -36 \\ 0 & 1 & \frac{17}{2} & 40 \\ 0 & 0 & \frac{105}{2} & 294 \end{bmatrix}$$ --- #### **Step 2.4:** Make the pivot element in row 3, column 3, equal to 1. - Divide $$R_3$$ by $$\frac{105}{2}$$: $$R_3 \rightarrow \frac{R_3}{\frac{105}{2}}$$ New matrix: $$\begin{bmatrix} 1 & 0 & -11 & -36 \\ 0 & 1 & \frac{17}{2} & 40 \\ 0 & 0 & 1 & 5.6 \end{bmatrix}$$ --- #### **Step 2.5:** Use $$R_3$$ to eliminate other entries in column 3. - Add $$11R_3$$ to $$R_1$$: $$R_1 \rightarrow R_1 + 11R_3$$ - Subtract $$\frac{17}{2}R_3$$ from $$R_2$$: $$R_2 \rightarrow R_2 - \frac{17}{2}R_3$$ Final matrix: $$\begin{bmatrix} 1 & 0 & 0 & 25.6 \\ 0 & 1 & 0 & 2.8 \\ 0 & 0 & 1 & 5.6 \end{bmatrix}$$ --- ### Step 3: Write the solution The solution is: $$x = 25.6, \quad y = 2.8, \quad z = 5.6$$ Would you like a detailed explanation of any specific step? --- ### Related Questions: 1. What are the main differences between Gauss-Jordan and Gaussian elimination? 2. How does pivoting improve numerical stability in solving systems of equations? 3. Can you explain how to check if the system of equations has no solution or infinite solutions? 4. How would the solution change if one equation were inconsistent with the others? 5. Can Gauss-Jordan elimination be used for non-square matrices? **Tip:** Always verify the solution by substituting the values back into the original equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Gauss-Jordan Elimination
Formulas
Row operations: Swap rows, Multiply row by a scalar, Add/subtract a multiple of one row to another
Reduced row-echelon form: Each leading entry in a row is 1, and all elements above and below are 0
Theorems
Gauss-Jordan Method for solving linear systems
Uniqueness of solutions for square matrices with non-zero determinant
Suitable Grade Level
Undergraduate or advanced high school (Grades 11-12)
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