Math Problem Statement
Use Gauss-Jordan row reduction to solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters x, y, and/or z.)
−x
+
2y
−
z
=
0
−x
−
y
+
2z
=
0
2x
−
z
=
6
(x, y, z) =
Solution
To solve this system using Gauss-Jordan elimination, we will write the system of equations in augmented matrix form and reduce it step by step to row-echelon form, followed by reduced row-echelon form.
The system of equations is:
Step 1: Write the augmented matrix
The augmented matrix for the system is:
-1 & 2 & -1 & 0 \\ -1 & -1 & 2 & 0 \\ 2 & 0 & -1 & 6 \end{bmatrix}$$ ### Step 2: Apply Gauss-Jordan elimination #### Step 2.1: Normalize the first row Divide the first row by $$-1$$ to make the pivot element $$1$$: $$\begin{bmatrix} 1 & -2 & 1 & 0 \\ -1 & -1 & 2 & 0 \\ 2 & 0 & -1 & 6 \end{bmatrix}$$ #### Step 2.2: Eliminate the first column below the pivot Add the first row to the second row and subtract $$2$$ times the first row from the third row: - $$R_2 = R_2 + R_1$$ - $$R_3 = R_3 - 2R_1$$ $$\begin{bmatrix} 1 & -2 & 1 & 0 \\ 0 & -3 & 3 & 0 \\ 0 & 4 & -3 & 6 \end{bmatrix}$$ #### Step 2.3: Normalize the second row Divide the second row by $$-3$$: $$\begin{bmatrix} 1 & -2 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 4 & -3 & 6 \end{bmatrix}$$ #### Step 2.4: Eliminate the second column above and below the pivot - $$R_1 = R_1 + 2R_2$$ - $$R_3 = R_3 - 4R_2$$ $$\begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 1 & 6 \end{bmatrix}$$ #### Step 2.5: Eliminate the third column above the pivot - $$R_1 = R_1 + R_3$$ - $$R_2 = R_2 + R_3$$ $$\begin{bmatrix} 1 & 0 & 0 & 6 \\ 0 & 1 & 0 & 6 \\ 0 & 0 & 1 & 6 \end{bmatrix}$$ ### Step 3: Interpret the result From the reduced matrix, we can write the solution: $$x = 6, \quad y = 6, \quad z = 6$$ Thus, the solution is: $$(x, y, z) = (6, 6, 6)$$ --- Would you like further details or have any questions? Here are 5 related questions for practice: 1. How does Gauss-Jordan elimination differ from Gaussian elimination? 2. What happens if the system is inconsistent? How would you identify that? 3. How can parametric solutions be expressed for dependent systems? 4. What are the advantages of using Gauss-Jordan elimination over substitution methods? 5. How would you handle a 4x4 system using this method? **Tip:** Always check for row swaps if a pivot element is zero to avoid errors in row reduction!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Operations
Formulas
Augmented matrix representation
Row reduction for Gauss-Jordan elimination
Theorems
Gauss-Jordan elimination process
Uniqueness of reduced row-echelon form
Suitable Grade Level
Grades 10-12
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