Math Problem Statement

Solve the following linear system of equations using the Gauss-Jordan method: - x₁ + 3x₂ - 10x₃ - 4x₄ - 4x₅ = 1 3x₁ - x₂ + 5x₄ - 2x₅ = -2 2x₁ + 2x₂ - 19x₃ + x₄ - 6x₅ = 5

Solution

The image you have uploaded contains a system of linear equations. The task is to solve this system using the Gauss-Jordan method. Here is the system extracted from the image:

1. \quad & -x_1 + 3x_2 - 10x_3 - 4x_4 - 4x_5 = 1 \\ 2. \quad & 3x_1 - x_2 + 5x_4 - 2x_5 = -2 \\ 3. \quad & 2x_1 + 2x_2 - 19x_3 + x_4 - 6x_5 = 5 \end{aligned}$$ I will now solve this system of equations using the Gauss-Jordan elimination method, transforming the augmented matrix into its reduced row-echelon form (RREF). Let me calculate this step-by-step.The reduced row-echelon form (RREF) of the augmented matrix is: $$\begin{pmatrix} 1 & 0 & 0 & 1.375 & -1.25 & -1.458 \\ 0 & 1 & 0 & -0.875 & -1.75 & -2.375 \\ 0 & 0 & 1 & 0 & 0 & -0.667 \end{pmatrix}$$ From this, we can extract the solutions for $$x_1$$, $$x_2$$, and $$x_3$$: $$x_1 = 1.375x_4 - 1.25x_5 - 1.458$$ $$x_2 = -0.875x_4 - 1.75x_5 - 2.375$$ $$x_3 = -0.667$$ Thus, $$x_3$$ has a fixed value, while $$x_1$$ and $$x_2$$ are dependent on $$x_4$$ and $$x_5$$, which are free variables. Would you like more details or clarifications on any step? Here are 5 related questions: 1. What are free variables in a system of equations? 2. How does the Gauss-Jordan method differ from Gaussian elimination? 3. What is the significance of the RREF in solving systems of equations? 4. Can all systems of linear equations be solved using Gauss-Jordan elimination? 5. How do you interpret solutions in terms of parametric form? **Tip:** When solving systems with free variables, solutions are often written in parametric form to express the dependency on the free variables.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Gauss-Jordan Elimination
Row Echelon Form

Formulas

Gaussian elimination steps
Row operations
Reduced Row Echelon Form (RREF)

Theorems

Gauss-Jordan Theorem
Properties of Linear Equations
Matrix Operations

Suitable Grade Level

Undergraduate Mathematics or Linear Algebra Course

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