Solving a System of Linear Equations Using Gauss-Jordan Elimination

Math Problem Statement

Tentukan solusi x, y, dan z dari sistem persamaan linear menggunakan Metode Gauss-Jordan: 1. x - 3y + z = 4 2. 2x - y = -2 3. 4x - 3z = 0

The problem in the image asks to solve a system of linear equations using the Gauss-Jordan method. The system of equations is:

x - 3y + z &= 4 \\ 2x - y &= -2 \\ 4x - 3z &= 0 \end{aligned}$$ Let's use the Gauss-Jordan elimination to solve for $$x$$, $$y$$, and $$z$$. ### Step 1: Write the augmented matrix for the system The augmented matrix for the system is: $$\begin{pmatrix} 1 & -3 & 1 & | & 4 \\ 2 & -1 & 0 & | & -2 \\ 4 & 0 & -3 & | & 0 \end{pmatrix}$$ ### Step 2: Apply the Gauss-Jordan elimination I'll now solve this system using the Gauss-Jordan method.After applying the Gauss-Jordan elimination, the resulting augmented matrix is: $$\begin{pmatrix} 1 & 0 & 0 & | & -2.73 \\ 0 & 1 & 0 & | & -3.45 \\ 0 & 0 & 1 & | & -3.64 \end{pmatrix}$$ From this matrix, we can directly read the solution for $$x$$, $$y$$, and $$z$$: $$x = -2.73, \quad y = -3.45, \quad z = -3.64$$ Would you like further details on this process or have any questions? ### Related questions: 1. What is the difference between Gaussian elimination and Gauss-Jordan elimination? 2. How does the Gauss-Jordan method handle systems with no solutions? 3. Can the Gauss-Jordan method be used for non-square matrices? 4. How does row-swapping affect the Gauss-Jordan elimination process? 5. How do you identify if a system has infinitely many solutions using Gauss-Jordan? ### Tip: Always check the determinant of the coefficient matrix in square systems to know whether the system has a unique solution (non-zero determinant).

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Linear Algebra
Gauss-Jordan Elimination
Systems of Linear Equations

Gauss-Jordan Elimination Matrix Form

Gauss-Jordan Method

Grade 10-12

Solving a System of Equations using Gauss-Jordan Elimination

Solving Systems of Equations Using Gauss-Jordan Elimination

Solving Systems of Linear Equations using Gauss-Jordan Method

Solving Systems of Linear Equations Using Gauss-Jordan Elimination

Solving System of Equations Using Gauss Elimination Method: 3x - y + 2z = 12, x + 2y + 8z = 11, 2x - 2y - z = 2