Math Problem Statement
Solve gauss elimination method 3x+2y=4 2x-y=5
Solution
To solve the system of linear equations using Gaussian elimination, we follow these steps:
System of Equations:
Write as an Augmented Matrix:
The system is represented as:
3 & 2 & | & 4 \\ 2 & -1 & | & 5 \end{bmatrix}$$ ### Step 1: Elimination to Form Row Echelon Matrix #### Eliminate $$x$$ from the second row: - Multiply Row 1 by $$\frac{2}{3}$$, then subtract it from Row 2: $$R_2 \rightarrow R_2 - \frac{2}{3} R_1$$ The calculations are as follows: - Row 1 remains unchanged: $$[3, 2, 4]$$ - New Row 2: $$2 - \frac{2}{3}(3) = 0, \quad -1 - \frac{2}{3}(2) = -\frac{7}{3}, \quad 5 - \frac{2}{3}(4) = \frac{7}{3}$$ Resulting matrix: $$\begin{bmatrix} 3 & 2 & | & 4 \\ 0 & -\frac{7}{3} & | & \frac{7}{3} \end{bmatrix}$$ #### Simplify Row 2: - Divide Row 2 by $$-\frac{7}{3}$$: $$R_2 \rightarrow R_2 \div -\frac{7}{3}$$ New Row 2: $$[0, 1, |, -1]$$ Resulting matrix: $$\begin{bmatrix} 3 & 2 & | & 4 \\ 0 & 1 & | & -1 \end{bmatrix}$$ #### Eliminate $$y$$ from Row 1: - Subtract $$2 \times R_2$$ from Row 1: $$R_1 \rightarrow R_1 - 2 \cdot R_2$$ The calculations are as follows: - New Row 1: $$3, \quad 2 - 2(1) = 0, \quad 4 - 2(-1) = 6$$ Resulting matrix: $$\begin{bmatrix} 3 & 0 & | & 6 \\ 0 & 1 & | & -1 \end{bmatrix}$$ ### Step 2: Back-Substitution From the Row Echelon Matrix: 1. From Row 2: $$y = -1$$ 2. From Row 1: $$3x = 6 \implies x = 2$$ ### Solution: $$x = 2, \, y = -1$$ Let me know if you need further clarification! --- ### Expanding Your Understanding 1. What is the difference between Gaussian elimination and Gauss-Jordan elimination? 2. How can pivoting improve the accuracy of Gaussian elimination? 3. Can Gaussian elimination work for any $$n \times n$$ system? 4. What happens if the determinant of the coefficient matrix is zero? 5. Why do we convert to row-echelon form before solving? ### Tip: When performing Gaussian elimination, always double-check arithmetic during elimination and pivoting to avoid computational errors!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Systems of Linear Equations
Formulas
Gaussian Elimination Algorithm
Theorems
Row Echelon Form
Back-Substitution
Suitable Grade Level
Grades 9-12
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