Math Problem Statement
TAE-2 - Div- II.pdf
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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
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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
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