Math Problem Statement
how to solve this one using the gaussian elimination?
How to solve this one using Gaussian elimination?
Solution
To solve this system of linear equations using Gaussian elimination, the goal is to convert the system into an upper triangular form, then back-substitute to find the values of , , and .
The system of equations is:
1) & \quad 2x - 3y - 9z = -5 \\ 2) & \quad x + 3z = 2 \\ 3) & \quad -3x + y - 4z = -3 \end{aligned}$$ ### Step 1: Write the augmented matrix for the system. $$\begin{pmatrix} 2 & -3 & -9 & | & -5 \\ 1 & 0 & 3 & | & 2 \\ -3 & 1 & -4 & | & -3 \end{pmatrix}$$ ### Step 2: Perform row operations to get an upper triangular matrix. We aim to create zeros below the diagonal. #### a) Make the first element of row 2 zero: We already have a 1 in the second row, first column (i.e., in row 2, column 1), so we can eliminate the first element in row 1. Use **Row 1 = Row 1 - 2 × Row 2**. $$\text{Row 1: } \begin{pmatrix} 2 & -3 & -9 & | & -5 \end{pmatrix} - 2 \times \begin{pmatrix} 1 & 0 & 3 & | & 2 \end{pmatrix}$$ New row 1: $$\begin{pmatrix} 0 & -3 & -15 & | & -9 \end{pmatrix}$$ The matrix now becomes: $$\begin{pmatrix} 0 & -3 & -15 & | & -9 \\ 1 & 0 & 3 & | & 2 \\ -3 & 1 & -4 & | & -3 \end{pmatrix}$$ #### b) Eliminate the first element in row 3: Use **Row 3 = Row 3 + 3 × Row 2** to eliminate the $$-3$$ in the first column of row 3. $$\text{Row 3: } \begin{pmatrix} -3 & 1 & -4 & | & -3 \end{pmatrix} + 3 \times \begin{pmatrix} 1 & 0 & 3 & | & 2 \end{pmatrix}$$ New row 3: $$\begin{pmatrix} 0 & 1 & 5 & | & 3 \end{pmatrix}$$ The updated matrix becomes: $$\begin{pmatrix} 0 & -3 & -15 & | & -9 \\ 1 & 0 & 3 & | & 2 \\ 0 & 1 & 5 & | & 3 \end{pmatrix}$$ ### Step 3: Scale the second row: To make the leading coefficient in the second row equal to 1, divide the entire second row by $$-3$$. $$\text{Row 1: } \begin{pmatrix} 0 & -3 & -15 & | & -9 \end{pmatrix} \Rightarrow \begin{pmatrix} 0 & 1 & 5 & | & 3 \end{pmatrix}$$ Updated matrix:Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Gaussian Elimination
Formulas
Gaussian Elimination Matrix Transformations
Row Operations
Theorems
Gaussian Elimination
Suitable Grade Level
College level or advanced high school