Math Problem Statement
Solution
Let's begin by solving the system of equations using the Gauss-Jordan method:
The system of equations is:
Step 1: Convert the system into an augmented matrix.
Rewriting the system in standard form:
x - y & = 3 & \text{(from equation 1)} \\ y - z & = 4 & \text{(from equation 2)} \\ x + z & = -1 & \text{(from equation 3)} \end{aligned}$$ This corresponds to the augmented matrix: $$\begin{pmatrix} 1 & -1 & 0 & | & 3 \\ 0 & 1 & -1 & | & 4 \\ 1 & 0 & 1 & | & -1 \end{pmatrix}$$ ### Step 2: Perform row operations to achieve reduced row-echelon form (RREF). 1. Subtract Row 1 from Row 3 to eliminate the $$x$$-term in Row 3: $$R_3 = R_3 - R_1 \quad \Rightarrow \quad \begin{pmatrix} 1 & -1 & 0 & | & 3 \\ 0 & 1 & -1 & | & 4 \\ 0 & 1 & 1 & | & -4 \end{pmatrix}$$ 2. Subtract Row 2 from Row 3 to simplify: $$R_3 = R_3 - R_2 \quad \Rightarrow \quad \begin{pmatrix} 1 & -1 & 0 & | & 3 \\ 0 & 1 & -1 & | & 4 \\ 0 & 0 & 2 & | & -8 \end{pmatrix}$$ 3. Divide Row 3 by 2 to normalize: $$R_3 = \frac{R_3}{2} \quad \Rightarrow \quad \begin{pmatrix} 1 & -1 & 0 & | & 3 \\ 0 & 1 & -1 & | & 4 \\ 0 & 0 & 1 & | & -4 \end{pmatrix}$$ 4. Use Row 3 to eliminate the $$z$$-terms from Row 2: $$R_2 = R_2 + R_3 \quad \Rightarrow \quad \begin{pmatrix} 1 & -1 & 0 & | & 3 \\ 0 & 1 & 0 & | & 0 \\ 0 & 0 & 1 & | & -4 \end{pmatrix}$$ 5. Use Row 2 to eliminate the $$y$$-term in Row 1: $$R_1 = R_1 + R_2 \quad \Rightarrow \quad \begin{pmatrix} 1 & 0 & 0 & | & 3 \\ 0 & 1 & 0 & | & 0 \\ 0 & 0 & 1 & | & -4 \end{pmatrix}$$ ### Step 3: Read the solution from the matrix. The resulting matrix gives the system: $$\begin{aligned} x & = 3 \\ y & = 0 \\ z & = -4 \end{aligned}$$ Thus, the solution is $$(x, y, z) = (3, 0, -4)$$. ### Conclusion: The correct choice is **A**: There is one solution, $$(3, 0, -4)$$, in the order $$x, y, z$$. --- Would you like more details on any step or further clarification? Here are five related questions you might find helpful: 1. How do you convert a system of equations into an augmented matrix? 2. What is the Gauss-Jordan elimination method, and how does it differ from Gaussian elimination? 3. Can a system of equations have no solution or infinitely many solutions? 4. How can you identify if a system is consistent or inconsistent? 5. How does the reduced row-echelon form help in solving systems of linear equations? **Tip:** Always double-check your row operations during elimination to avoid small arithmetic errors that can lead to incorrect solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Gauss-Jordan Elimination
Formulas
Gauss-Jordan Elimination Matrix Method
Theorems
Gauss-Jordan Elimination Theorem
Suitable Grade Level
Grades 10-12
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