Math Problem Statement
solve it
Solution
To solve the given system using the Gauss-Jordan Elimination method, we will start by converting the system of equations into an augmented matrix and then perform row operations to get the system in reduced row echelon form.
Given system of equations:
Step 1: Write the augmented matrix
The augmented matrix for the system is:
-2 & 6 & -4 & -28 \\ -1 & 3 & -1 & -8 \\ 5 & -15 & 10 & 70 \\ 1 & -3 & 0 & 2 \end{bmatrix}$$ ### Step 2: Perform Gauss-Jordan elimination We will now perform row operations to simplify this matrix to reduced row echelon form (RREF). Let's perform the necessary row operations: 1. First, we'll swap row 1 and row 4 to get a leading 1 in the top-left corner. $$\begin{bmatrix} 1 & -3 & 0 & 2 \\ -1 & 3 & -1 & -8 \\ 5 & -15 & 10 & 70 \\ -2 & 6 & -4 & -28 \end{bmatrix}$$ 2. Next, we'll eliminate the $$x$$-terms in rows 2, 3, and 4 using row 1. Add row 1 to row 2, subtract 5 times row 1 from row 3, and add 2 times row 1 to row 4. After performing these operations, we get the matrix: $$\begin{bmatrix} 1 & -3 & 0 & 2 \\ 0 & 0 & -1 & -6 \\ 0 & 0 & 10 & 60 \\ 0 & 0 & -4 & -24 \end{bmatrix}$$ 3. Now, we want to deal with the $$z$$-terms. We’ll simplify row 3 by dividing it by 10 and row 4 by -4. After these steps: $$\begin{bmatrix} 1 & -3 & 0 & 2 \\ 0 & 0 & -1 & -6 \\ 0 & 0 & 1 & 6 \\ 0 & 0 & 1 & 6 \end{bmatrix}$$ 4. Now we can eliminate the $$z$$-terms in rows 1 and 2 by adding row 3 to row 2 and subtracting row 3 from row 4. The matrix becomes: $$\begin{bmatrix} 1 & -3 & 0 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ ### Step 3: Read the solution from the matrix From this final matrix, we can see that: - $$x - 3y = 2$$ (from the first row). - $$z = 6$$ (from the third row). Substitute $$z = 6$$ into the equation $$x - 3y = 2$$: $$x - 3y = 2 \quad \text{and} \quad z = 6.$$ This system can be solved for $$x$$ and $$y$$ using additional steps (such as substitution), but it looks like the system has infinite solutions, as we have free variables $$y$$. Would you like to go deeper into the solution process, or do you have any specific questions about it? Here are a few related questions that could help expand your understanding: 1. How do you perform the Gauss-Jordan elimination step-by-step in detail? 2. What happens when a system has infinitely many solutions, and how do you express those solutions? 3. How do you interpret a row of zeros in the augmented matrix? 4. Can Gauss-Jordan elimination be applied to systems that are inconsistent? 5. How do you perform Gauss-Jordan elimination on a system of nonlinear equations? **Tip:** Always check for consistency in the augmented matrix—if you find a row where all coefficients are zero but the constant is non-zero, the system has no solution!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Operations
Formulas
Gauss-Jordan Elimination steps for row reduction
Theorems
Properties of row-echelon and reduced row-echelon matrices
Suitable Grade Level
Undergraduate or Advanced High School
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