Math Problem Statement

Write the system of equations corresponding to the augmented matrix. Perform row operations R2 = -2R1 + R2 and R3 = R1 + R3 on the given augmented matrix.

Solution

To solve this problem, let's analyze the augmented matrix and the associated system of equations step by step.

Given Augmented Matrix:

-1 & 2 & 3 & 3 \\ 2 & -5 & 6 & 5 \\ -7 & 5 & 3 & 5 \end{bmatrix}$$ The columns represent the coefficients of $$x$$, $$y$$, and $$z$$, with the rightmost column being the constants on the right-hand side of the equations. ### Step 1: Write the corresponding system of equations From the augmented matrix, we can directly extract the system of equations: 1. $$-x + 2y + 3z = 3$$ 2. $$2x - 5y + 6z = 5$$ 3. $$-7x + 5y + 3z = 5$$ This matches option **D**. ### Step 2: Perform the given row operations The operations are: 1. $$R_2 \to -2R_1 + R_2$$ 2. $$R_3 \to R_1 + R_3$$ #### Row operation 1: $$R_2 \to -2R_1 + R_2$$ $$R_2 = (-2)(-1, 2, 3, 3) + (2, -5, 6, 5)$$ $$R_2 = (2, -4, -6, -6) + (2, -5, 6, 5) = (4, -9, 0, -1)$$ Updated matrix: $$\begin{bmatrix} -1 & 2 & 3 & 3 \\ 4 & -9 & 0 & -1 \\ -7 & 5 & 3 & 5 \end{bmatrix}$$ #### Row operation 2: $$R_3 \to R_1 + R_3$$ $$R_3 = (-1, 2, 3, 3) + (-7, 5, 3, 5)$$ $$R_3 = (-8, 7, 6, 8)$$ Updated matrix: $$\begin{bmatrix} -1 & 2 & 3 & 3 \\ 4 & -9 & 0 & -1 \\ -8 & 7 & 6 & 8 \end{bmatrix}$$ The transformed matrix corresponds to a new system of equations, but based on the original system provided, option **D** is correct as it aligns with the initial augmented matrix. --- Would you like further clarification on the row operations or the augmented matrix? Here are some follow-up questions: 1. What are augmented matrices and how do they relate to systems of equations? 2. How do row operations affect the solutions of a system of equations? 3. Can you explain how to interpret augmented matrices in general? 4. How do you identify if a system is consistent or inconsistent from its augmented matrix? 5. How do row reduction techniques help solve systems of equations? **Tip:** When performing row operations, carefully track each step to ensure accuracy in updating the matrix values.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Operations

Formulas

Row Operation Formula
Matrix Representation of Systems of Equations

Theorems

Consistency of Linear Systems
Gaussian Elimination

Suitable Grade Level

Grades 10-12 or College Level