Math Problem Statement

Find x

Find the values of x in the system of linear equations.

Solution

The image you provided shows a system of linear equations that looks like this:

-1 & 1 & 0 \\ 0 & -1 & 4 \\ -4 & 0 & -25 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$$ This is a homogeneous system of equations, and we want to find the values of $$x_1$$, $$x_2$$, and $$x_3$$. To solve this, we can set up the system of equations from the matrix: 1. From the first row: $$-x_1 + x_2 = 0$$ 2. From the second row: $$-x_2 + 4x_3 = 0$$ 3. From the third row: $$-4x_1 - 25x_3 = 0$$ Let's solve this step by step: ### Step 1: Solve for $$x_1$$ and $$x_2$$ from the first equation From $$-x_1 + x_2 = 0$$, we get: $$x_1 = x_2$$ ### Step 2: Solve for $$x_2$$ and $$x_3$$ from the second equation From $$-x_2 + 4x_3 = 0$$, we get: $$x_2 = 4x_3$$ ### Step 3: Substitute into the third equation Substitute $$x_1 = 4x_3$$ into the third equation: $$-4x_1 - 25x_3 = 0$$ $$-4(4x_3) - 25x_3 = 0$$ $$-16x_3 - 25x_3 = 0$$ $$-41x_3 = 0$$ Thus, $$x_3 = 0$$. ### Step 4: Solve for $$x_1$$ and $$x_2$$ Since $$x_3 = 0$$, from the previous relations: $$x_2 = 4x_3 = 0$$ $$x_1 = x_2 = 0$$ ### Conclusion: The solution to the system is $$x_1 = x_2 = x_3 = 0$$. This system has only the trivial solution, meaning all variables are zero. Would you like any more details on this, or do you have further questions? Here are five related questions you might want to explore: 1. How can we identify whether a homogeneous system has a non-trivial solution? 2. What is the geometric interpretation of the solution to a homogeneous system? 3. How can we solve non-homogeneous systems of linear equations? 4. What role does the determinant of the coefficient matrix play in solving such systems? 5. How does matrix rank relate to the solution of linear systems? **Tip**: For any system of equations, the determinant of the matrix can help determine whether the system has unique, infinite, or no solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Equations
Homogeneous Systems

Formulas

Matrix multiplication
Solving systems of linear equations

Theorems

Homogeneous system theorem
Gauss-Jordan elimination
Rank-nullity theorem

Suitable Grade Level

Undergraduate (Linear Algebra)

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