Math Problem Statement

Solve

Solve the system using row operations (or elementary matrices): 2x + 3y + 5z = -11, 5x + 2y + 2z = -13, 3x + 3y + 6z = -18.

Solution

To solve the given system of linear equations using row operations or elementary matrices, let's rewrite the problem in an augmented matrix form and solve step-by-step.

The system is:

2x + 3y + 5z &= -11, \\ 5x + 2y + 2z &= -13, \\ 3x + 3y + 6z &= -18. \end{aligned}$$ The corresponding augmented matrix is: $$\begin{bmatrix} 2 & 3 & 5 & | & -11 \\ 5 & 2 & 2 & | & -13 \\ 3 & 3 & 6 & | & -18 \end{bmatrix}.$$ Now, we'll perform row operations to bring this matrix to reduced row-echelon form. sympy as sp # Define the augmented matrix augmented_matrix = sp.Matrix([ [2, 3, 5, -11], [5, 2, 2, -13], [3, 3, 6, -18] ]) # Perform row reduction to reduced row-echelon form (RREF) rref_matrix, pivots = augmented_matrix.rref() rref_matrixThe system of equations reduces to the following in row-echelon form: $$\begin{bmatrix} 1 & 0 & 0 & | & -3 \\ 0 & 1 & 0 & | & 5 \\ 0 & 0 & 1 & | & -4 \end{bmatrix}.$$ This corresponds to: $$x = -3, \quad y = 5, \quad z = -4.$$ ### Final Solution: $$x = -3, \, y = 5, \, z = -4$$ Let me know if you'd like a detailed step-by-step breakdown of the row operations used. ### Follow-Up Questions: 1. What are the advantages of using row operations versus substitution for solving systems of equations? 2. How can you verify this solution is correct? 3. Can every system of linear equations be solved using row operations? 4. What happens when a system has infinitely many solutions? How does the row-reduction process indicate this? 5. Can elementary matrices be used in this problem instead of standard row operations? **Tip:** Practicing row-reduction by hand helps you understand how to identify dependent systems and inconsistencies.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Row Operations

Formulas

Row-reduction (Gaussian elimination)
Matrix representation of linear systems

Theorems

Gaussian Elimination
Elementary Matrix Theorem

Suitable Grade Level

Undergraduate (Linear Algebra)