Solving System of Linear Equations using Gaussian Elimination

Math Problem Statement

Given system of linear equations: x - 4y + z = 0, 2x + y + 4z = 0, x + 4y + z = 0, have:

Solution

Let's analyze the system of linear equations given in the image:

1. \ & x - 4y + z = 0 \\ 2. \ & 2x + y + 4z = 0 \\ 3. \ & x + 4y + z = 0 \\ \end{aligned}$$ To determine the type of solution (trivial, non-trivial, no solution, or $$n-r$$ solutions), we can proceed by solving this system. ### Step 1: Convert the system into matrix form The augmented matrix for this system is: $$\begin{pmatrix} 1 & -4 & 1 & | & 0 \\ 2 & 1 & 4 & | & 0 \\ 1 & 4 & 1 & | & 0 \\ \end{pmatrix}$$ ### Step 2: Perform Gaussian elimination Start by making the first pivot entry in row 1 equal to 1 (which it already is). Now, eliminate the first column entries in rows 2 and 3. - Subtract 2 times row 1 from row 2: $$(R_2 - 2R_1) \to R_2$$ Row 2 becomes: $$\begin{pmatrix} 0 & 9 & 2 & | & 0 \end{pmatrix}$$ - Subtract row 1 from row 3: $$(R_3 - R_1) \to R_3$$ Row 3 becomes: $$\begin{pmatrix} 0 & 8 & 0 & | & 0 \end{pmatrix}$$ Now, the matrix is: $$\begin{pmatrix} 1 & -4 & 1 & | & 0 \\ 0 & 9 & 2 & | & 0 \\ 0 & 8 & 0 & | & 0 \\ \end{pmatrix}$$ ### Step 3: Further eliminate and solve Next, we can work on row 3 by eliminating the second column: - Subtract $$\frac{8}{9} \times \text{Row 2}$$ from Row 3 to eliminate the second entry in row 3: $$R_3 - \frac{8}{9}R_2 \to R_3$$ Row 3 becomes: $$\begin{pmatrix} 0 & 0 & -\frac{16}{9} & | & 0 \end{pmatrix}$$ The matrix now is: $$\begin{pmatrix} 1 & -4 & 1 & | & 0 \\ 0 & 9 & 2 & | & 0 \\ 0 & 0 & -\frac{16}{9} & | & 0 \\ \end{pmatrix}$$ ### Step 4: Analyze the system We now have a consistent system, with no contradictory equations. The third row indicates that $$z = 0$$. Substituting into the second row, we find $$y = 0$$, and substituting into the first row, we find $$x = 0$$. Thus, the only solution is the trivial solution: $$x = 0, y = 0, z = 0$$. ### Conclusion: The system has **only the trivial solution**. Therefore, the correct answer is: **A. trivial solution** --- Would you like more details on any step? Here are some related questions for further practice: 1. How would the solution change if the third equation were different? 2. Can you explain the significance of trivial vs. non-trivial solutions? 3. What is the geometric interpretation of this system of equations? 4. How would you determine if a system has infinitely many solutions? 5. What is the role of Gaussian elimination in solving systems of linear equations? **Tip:** Always check if the matrix is consistent by reducing it fully and looking for contradictions!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Gaussian Elimination

Formulas

Augmented matrix of system
Row operations for Gaussian elimination

Theorems

Gaussian elimination method
Uniqueness of solutions for consistent systems

Suitable Grade Level

Grades 10-12 or early college

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