Math Problem Statement
Selesaikan sistem persamaan linear berikut menggunakan metode eliminasi Gauss:
1. 2x₁ + 3x₂ - x₃ = 5
2. 4x₁ + x₂ + 2x₃ = 11
3. -2x₁ + 5x₂ + 3x₃ = 0
Solution
The system of linear equations displayed in the image is as follows:
2x_1 + 3x_2 - x_3 &= 5 \quad (1) \\ 4x_1 + x_2 + 2x_3 &= 11 \quad (2) \\ -2x_1 + 5x_2 + 3x_3 &= 0 \quad (3) \end{aligned}$$ The task is to solve this system using **Gaussian elimination**. Let's walk through the steps: ### Step 1: Form the augmented matrix The augmented matrix for this system is: $$\begin{pmatrix} 2 & 3 & -1 & | & 5 \\ 4 & 1 & 2 & | & 11 \\ -2 & 5 & 3 & | & 0 \end{pmatrix}$$ ### Step 2: Perform row operations to get the matrix in upper triangular form #### First, make the leading coefficient of the first row a 1 by dividing Row 1 by 2: $$\text{Row 1} \rightarrow \frac{1}{2} \text{Row 1}$$ $$\begin{pmatrix} 1 & 1.5 & -0.5 & | & 2.5 \\ 4 & 1 & 2 & | & 11 \\ -2 & 5 & 3 & | & 0 \end{pmatrix}$$ #### Second, eliminate the first element in Row 2 and Row 3: - Row 2: $$\text{Row 2} \rightarrow \text{Row 2} - 4 \times \text{Row 1}$$ - Row 3: $$\text{Row 3} \rightarrow \text{Row 3} + 2 \times \text{Row 1}$$ $$\begin{pmatrix} 1 & 1.5 & -0.5 & | & 2.5 \\ 0 & -5 & 4 & | & 1 \\ 0 & 8 & 2 & | & 5 \end{pmatrix}$$ ### Step 3: Eliminate the second element in Row 3: - Row 3: $$\text{Row 3} \rightarrow \text{Row 3} + \frac{8}{-5} \times \text{Row 2}$$ $$\begin{pmatrix} 1 & 1.5 & -0.5 & | & 2.5 \\ 0 & -5 & 4 & | & 1 \\ 0 & 0 & \frac{34}{5} & | & \frac{21}{5} \end{pmatrix}$$ ### Step 4: Back-substitution Now that we have an upper triangular matrix, we can solve the system by back-substitution: 1. From the third row: $$\frac{34}{5} x_3 = \frac{21}{5} \implies x_3 = \frac{21}{34} = 0.618$$ 2. From the second row: $$-5x_2 + 4x_3 = 1 \implies -5x_2 + 4(0.618) = 1 \implies -5x_2 = 1 - 2.472 = -1.472 \implies x_2 = 0.294$$ 3. From the first row: $$x_1 + 1.5x_2 - 0.5x_3 = 2.5 \implies x_1 + 1.5(0.294) - 0.5(0.618) = 2.5$$ Simplifying: $$x_1 + 0.441 - 0.309 = 2.5 \implies x_1 = 2.5 - 0.132 = 2.368$$ ### Final Solution: $$x_1 = 2.368, \quad x_2 = 0.294, \quad x_3 = 0.618$$ Would you like further details on any step? Here are some follow-up questions: 1. How can you verify that the solution is correct? 2. What is Gaussian elimination used for in general? 3. What if the system had infinitely many solutions? How would we detect that? 4. Can this method be applied to larger systems of equations? 5. How would the process differ for a 2x2 system? **Tip:** When solving systems of linear equations using Gaussian elimination, it's important to keep track of your row operations to avoid calculation 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 method
Row operations for elimination
Back-substitution in upper triangular form
Theorems
Gaussian Elimination Theorem
Suitable Grade Level
Grades 10-12
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