Math Problem Statement
Use the Gauss-Jordan method to solve the system of equations: x + 2y + z = -2, 2x + y - z = 1, x + y - 2z = -3.
Solution
To solve the given system of equations using the Gauss-Jordan method, let's rewrite it in augmented matrix form and then reduce it to row-reduced echelon form (RREF):
System of equations:
Augmented matrix:
1 & 2 & 1 & -2 \\ 2 & 1 & -1 & 1 \\ 1 & 1 & -2 & -3 \end{bmatrix}$$ Now, we proceed with the Gauss-Jordan elimination: 1. **Step 1: Make the first pivot (top-left) equal to 1.** Divide the first row by 1 (it's already 1): $$\begin{bmatrix} 1 & 2 & 1 & -2 \\ 2 & 1 & -1 & 1 \\ 1 & 1 & -2 & -3 \end{bmatrix}$$ 2. **Step 2: Eliminate the first column for rows 2 and 3.** - For row 2: $$R_2 \to R_2 - 2R_1$$ - For row 3: $$R_3 \to R_3 - R_1$$ $$\begin{bmatrix} 1 & 2 & 1 & -2 \\ 0 & -3 & -3 & 5 \\ 0 & -1 & -3 & -1 \end{bmatrix}$$ 3. **Step 3: Make the pivot in the second column equal to 1.** Divide the second row by $$-3$$: $$\begin{bmatrix} 1 & 2 & 1 & -2 \\ 0 & 1 & 1 & -\frac{5}{3} \\ 0 & -1 & -3 & -1 \end{bmatrix}$$ 4. **Step 4: Eliminate the second column for rows 1 and 3.** - For row 1: $$R_1 \to R_1 - 2R_2$$ - For row 3: $$R_3 \to R_3 + R_2$$ $$\begin{bmatrix} 1 & 0 & -1 & \frac{4}{3} \\ 0 & 1 & 1 & -\frac{5}{3} \\ 0 & 0 & -2 & -\frac{8}{3} \end{bmatrix}$$ 5. **Step 5: Make the pivot in the third column equal to 1.** Divide the third row by $$-2$$: $$\begin{bmatrix} 1 & 0 & -1 & \frac{4}{3} \\ 0 & 1 & 1 & -\frac{5}{3} \\ 0 & 0 & 1 & \frac{4}{3} \end{bmatrix}$$ 6. **Step 6: Eliminate the third column for rows 1 and 2.** - For row 1: $$R_1 \to R_1 + R_3$$ - For row 2: $$R_2 \to R_2 - R_3$$ $$\begin{bmatrix} 1 & 0 & 0 & \frac{8}{3} \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & \frac{4}{3} \end{bmatrix}$$ ### Solution: From the row-reduced matrix, the solution is: $$x = \frac{8}{3}, \quad y = -3, \quad z = \frac{4}{3}.$$ Would you like further steps clarified or visualized? Here are five related questions: 1. Can the Gauss-Jordan method handle non-square systems of equations? 2. What is the significance of row-reduced echelon form? 3. How does this method compare to Gaussian elimination? 4. Could matrix inverses be used to solve the same system? 5. What happens if a system of equations has no solution or infinite solutions? **Tip:** Always check your final solution by substituting the variables back into the original equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrix Row Reduction
Formulas
Row Operations: Swapping, Scaling, and Row Replacement
Row-Reduced Echelon Form (RREF)
Theorems
Gauss-Jordan Elimination Theorem
Suitable Grade Level
Grades 10-12
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