Math Problem Statement

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Solve the system of linear homogeneous equations : x1+x2+x3+x4=0 x1+3x2+2x3+4x4=0 2x1+x3-x4=0

Solution

We are given a system of linear homogeneous equations:

x_1 + x_2 + x_3 + x_4 &= 0 \\ x_1 + 3x_2 + 2x_3 + 4x_4 &= 0 \\ 2x_1 + x_3 - x_4 &= 0 \end{aligned}$$ ### Step 1: Set up the augmented matrix We can express the system as an augmented matrix (without the right-hand side since it’s homogeneous): \[ \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 3 & 2 & 4 \\ 2 & 0 & 1 & -1

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Homogeneous Systems
Matrix Operations

Formulas

Augmented Matrix Setup
Row Reduction (Gaussian Elimination)

Theorems

Homogeneous System Solutions
Rank-Nullity Theorem

Suitable Grade Level

Grades 10-12