Math Problem Statement

TAE-2 - Div- II.pdf

205.47 KB

SOLVE QUESTION 5

Solution

To solve the system of linear equations:

3x + 2y - z = 1 \tag{1} x + 2y + 2z = 0 \tag{2} 2x + y + 3z = -1 \tag{3}

We can approach this problem using either substitution, elimination, or matrix methods. Let's use the matrix method, particularly Gaussian elimination, to solve this system.

Step 1: Set up the augmented matrix

The system of equations can be written as an augmented matrix:

3 & 2 & -1 & | & 1 \\ 1 & 2 & 2 & | & 0 \\ 2 & 1 & 3 & | & -1 \end{pmatrix}$$ ### Step 2: Perform row operations to get row echelon form 1. Make the pivot in the first column a 1 by swapping row 1 and row 2: $$\begin{pmatrix} 1 & 2 & 2 & | & 0 \\ 3 & 2 & -1 & | & 1 \\ 2 & 1 & 3 & | & -1 \end{pmatrix}$$ 2. Eliminate the entries below the pivot in the first column by replacing $$R_2$$ with $$R_2 - 3R_1$$ and $$R_3$$ with $$R_3 - 2R_1$$: $$\begin{pmatrix} 1 & 2 & 2 & | & 0 \\ 0 & -4 & -7 & | & 1 \\ 0 & -3 & -1 & | & -1 \end{pmatrix}$$ 3. Make the pivot in the second column a 1 by dividing $$R_2$$ by -4: \[ \begin{pmatrix} 1 & 2 & 2 & | & 0 \\ 0 & 1 & \frac{7}{4} & | & -\frac{1}{4} \\ 0 & -3 & -1 & | & -

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Matrix Operations

Formulas

Gaussian elimination
Matrix row operations

Theorems

Gaussian elimination method
Row echelon form

Suitable Grade Level

Grades 11-12 or early college