Math Problem Statement

Question content area top Part 1 Use the​ Gauss-Jordan method to solve the following system of equations. 3x plus 7y minus 2z equals 0 5x minus nothingy plus 5z equals 1 8x plus 6y plus 3z equals 1

Solution

To solve the system of equations using the Gauss-Jordan method, we'll first represent the system as an augmented matrix and then perform row operations to bring it into reduced row echelon form (RREF). The system of equations is:

1. & \ 3x + 7y - 2z = 0 \\ 2. & \ 5x + 0y + 5z = 1 \\ 3. & \ 8x + 6y + 3z = 1 \end{aligned}$$ ### Step 1: Write the augmented matrix $$\begin{bmatrix} 3 & 7 & -2 & | & 0 \\ 5 & 0 & 5 & | & 1 \\ 8 & 6 & 3 & | & 1 \end{bmatrix}$$ ### Step 2: Perform row operations to get the matrix into RREF I'll now perform the row operations necessary to solve this system.The system of equations has been transformed into the following reduced row echelon form (RREF): $$\begin{bmatrix} 1 & 0 & 0 & | & 0.08 \\ 0 & 1 & 0 & | & 0 \\ 0 & 0 & 1 & | & 0.12 \end{bmatrix}$$ From this, we can conclude the solution to the system of equations: $$\begin{aligned} x & = 0.08 \\ y & = 0 \\ z & = 0.12 \end{aligned}$$ ### Final Solution: $$x = 0.08, \quad y = 0, \quad z = 0.12$$ Would you like more details, or do you have any questions? Here are some related questions to deepen understanding: 1. How does the Gauss-Jordan method differ from Gaussian elimination? 2. What are the key steps in reducing a matrix to RREF? 3. How can row operations affect the solution of a system? 4. Can all systems of equations be solved using Gauss-Jordan elimination? 5. What are the advantages of using the Gauss-Jordan method? **Tip:** Always check if the system is consistent or inconsistent before starting the Gauss-Jordan method.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Gauss-Jordan Elimination
Augmented Matrix

Formulas

Gauss-Jordan elimination method for solving systems of linear equations
Row operations: swapping, multiplying, adding/subtracting rows

Theorems

Gauss-Jordan Elimination Theorem
Reduced Row Echelon Form (RREF)

Suitable Grade Level

Grades 9-12

Related Recommendation

Solving a System of Linear Equations Using Gauss-Jordan Method

Solve a System of Linear Equations using Gauss-Jordan Elimination

Solve a System of Linear Equations Using Gauss-Jordan Elimination

Solving Systems of Linear Equations using Gauss-Jordan Elimination: 2x + 2y + z = 4, x + z = 2, 4y − 3z = 12

Solving 3x3 Systems of Equations Using Gauss and Gauss-Jordan Elimination