Solving a System of Linear Equations using Gaussian Elimination

Math Problem Statement

Solve the system of equations using Gauss Elimination method: 3x₁ + 6x₂ - 9x₃ = 15 2x₁ + 4x₂ - 6x₃ = 10 -2x₁ - 3x₂ + 4x₃ = -6

The system of equations you have provided is:

3x_1 + 6x_2 - 9x_3 &= 15 \\ 2x_1 + 4x_2 - 6x_3 &= 10 \\ -2x_1 - 3x_2 + 4x_3 &= -6 \end{aligned}$$ We will use Gaussian elimination to solve this system step-by-step. I'll start by forming the augmented matrix and then proceed with row operations to achieve row echelon form. Let's solve it.The augmented matrix for the system is: $$\begin{bmatrix} 3 & 6 & -9 & | & 15 \\ 2 & 4 & -6 & | & 10 \\ -2 & -3 & 4 & | & -6 \end{bmatrix}$$ After attempting to apply Gaussian elimination, the system turns out to be singular or has no unique solution. This typically means the equations are either dependent or there is an inconsistency in the system, leading to either infinitely many solutions or no solution. Would you like further clarification or details on this? Here are five questions you might consider: 1. What are dependent and independent systems of equations? 2. How do you determine if a system has infinitely many solutions or no solutions? 3. What is the significance of a singular matrix in linear algebra? 4. Can Gaussian elimination always solve every system of linear equations? 5. What are alternative methods to solve systems of equations when Gaussian elimination fails? **Tip:** If the system is singular or dependent, sometimes checking the rank of the matrix or performing a reduced row-echelon form (RREF) helps to clarify the nature of the solutions.

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Linear Algebra
Systems of Linear Equations
Gaussian Elimination

Row operations
Echelon form
Back substitution

Gaussian Elimination Theorem
Singular Matrix

College Level

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