Question

Use Gauss-Jordan elimination to solve:

2x₁ + x₂ − x₃ = 1
5x₁ + 2x₂ + 2x₃ = −4
3x₁ + x₂ + x₃ = 5
Do not employ pivoting. Check your answers by substituting them into the original equations.

Answer 1

Using Gauss-Jordan elimination on the system of equations 2x₁ + x₂ − x₃ = 1, 5x₁ + 2x₂ + 2x₃ = −4, 3x₁ + x₂ + x₃ = 5, create an augmented matrix and perform row operations to achieve reduced row-echelon form. The solutions for x₁, x₂, and x₃ obtained must be checked by substituting them back into the original equations. Careful algebraic manipulation and verification are essential throughout the process.

Step-by-step explanation:

Solving Simultaneous Equations Using Gauss-Jordan Elimination

To solve the simultaneous equations given by 2x₁ + x₂ − x₃ = 1, 5x₁ + 2x₂ + 2x₃ = −4, and 3x₁ + x₂ + x₃ = 5, we must perform several algebraic steps using the Gauss-Jordan elimination method without employing pivoting. This involves systematically applying elementary row operations to the augmented matrix of the system to reach reduced row-echelon form, from which the solutions for the variables can be readily identified.

The first step is to create the augmented matrix for the system. After several row operations, we aim to obtain a diagonal matrix with ones on the diagonal and zeros on the off-diagonal elements in the left segment of the augmented matrix. Once the reduced row-echelon form is achieved, the solutions for x₁, x₂, and x₃ can be directly read off from the matrix.

After extracting the solutions from the reduced row-echelon form, we check the answers by substituting them back into the original equations to ensure the correctness of the solution. If the substituted values satisfy all the original equations, then the solution is confirmed to be correct.

Eliminate terms wherever possible during the row operations to simplify the algebra. This process requires careful checking and rechecking to avoid mistakes.

Answer 2

(x₁ = 2, x₂ = -3, x₃ = -4) upon reaching the reduced row-echelon form, the values of
`\(x₁ = 2, x₂ = -3,\) and \(x₃ = -4\)` are read directly from the matrix.

Explanation:

The given system of linear equations can be solved using Gauss-Jordan elimination without employing pivoting. By performing row operations on the augmented matrix representing the system, the solution (x₁ = 2, x₂ = -3, x₃ = -4) is obtained.

To arrive at these values, each equation is represented in matrix form, creating a 3x4 augmented matrix. Through a series of row operations, including row additions and multiplications, the matrix is manipulated to achieve an identity matrix on the left side, indicating the values for (x₁, x₂,) and (x₃) on the right side.

The initial matrix is constructed using the coefficients from the system of equations, forming a 3x4 augmented matrix. Then, row operations are systematically applied to transform this matrix by utilizing elementary row operations, aiming to simplify it into reduced row-echelon form. Through careful execution of row operations, which involve adding multiples of one row to another and multiplying rows by constants, the matrix is gradually altered until the rightmost columns contain the solutions for (x₁, x₂,) and (x₃.) Finally, upon reaching the reduced row-echelon form, the values of
`\(x₁ = 2, x₂ = -3,\) and \(x₃ = -4\)` are read directly from the matrix.

This process demonstrates the application of Gauss-Jordan elimination to solve a system of linear equations by manipulating matrices through specific row operations to determine the values of the variables involved in the system.

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