Math Problem Statement

Helena used row operations on an augmented matrix representing a system of equations and obtained the following result: [1 0 6] [0 0 0] How many solutions does Helena’s system of equations have? A. 0 B. 1 C. infinitely many D. cannot be determined

Solution

The augmented matrix Helena obtained is:

1 & 0 & 6 \\ 0 & 0 & 0 \end{bmatrix}$$ This matrix represents the system of linear equations after row operations. Let's interpret the rows: 1. The first row $$[1 \ 0 \ 6]$$ corresponds to the equation $$x_1 = 6$$. 2. The second row $$[0 \ 0 \ 0]$$ represents the equation $$0 = 0$$, which is always true and does not provide additional information. Given this, the system reduces to just $$x_1 = 6$$, and since there are no constraints on any other variables, the system has **infinitely many solutions** because at least one variable (e.g., $$x_2$$) is free. The correct answer is: - **C. Infinitely many solutions** Do you want more details or have any questions? Here are 5 questions related to this explanation: 1. What would happen if the second row had non-zero elements? 2. How do free variables affect the number of solutions? 3. What is a consistent system of linear equations? 4. How does row reduction lead to different forms of solutions? 5. What is the difference between no solution and infinitely many solutions? **Tip:** A row of zeros in an augmented matrix often indicates a free variable, leading to infinitely many solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Augmented Matrix
Systems of Equations
Row Echelon Form
Free Variables

Formulas

-

Theorems

Row Echelon Form
Existence and Uniqueness Theorem

Suitable Grade Level

Grades 10-12