Math Problem Statement

Consider the following augmented matrix. What can you say about the system of equations represented by this augmented matrix? A. It has no solutions. B. It has a unique solution. C. It has an infinite number of solutions. D. None of the above.

Solution

This question involves analyzing the augmented matrix for a system of linear equations. Let's examine the matrix given:

1 & -1 & 0 & 0 & | & 5 \\ 0 & 1 & 0 & 0 & | & 6 \\ 0 & 1 & -1 & 1 & | & 3 \\ 0 & 0 & 1 & -1 & | & 4 \\ \end{bmatrix}$$ Each row of this matrix represents a linear equation with four unknowns ($$x_1, x_2, x_3, x_4$$). The augmented matrix can be broken down into equations: 1. $$x_1 - x_2 = 5$$ 2. $$x_2 = 6$$ 3. $$x_2 - x_3 + x_4 = 3$$ 4. $$x_3 - x_4 = 4$$ ### Steps to Analyze the Solution 1. **Substitute $$x_2 = 6$$ from the second equation** into the others: - In the first equation, $$x_1 - 6 = 5 \Rightarrow x_1 = 11$$. 2. **Using $$x_2 = 6$$ in the third equation**: - $$6 - x_3 + x_4 = 3 \Rightarrow -x_3 + x_4 = -3 \Rightarrow x_3 - x_4 = 3$$. 3. **Compare with the fourth equation**: $$x_3 - x_4 = 4$$. - This results in a contradiction because $$x_3 - x_4$$ cannot be both 3 and 4 at the same time. ### Conclusion Due to the contradiction, this system of equations **has no solutions**. ### Answer The correct answer is: **A. It has no solutions.** Would you like more details on any part of this solution? --- Here are some related questions: 1. What is an augmented matrix and how is it used to represent systems of linear equations? 2. How do you determine if a system has no solution, a unique solution, or infinitely many solutions? 3. What are the steps to row reduce an augmented matrix to check for consistency? 4. How do contradictions in a system of equations lead to no solutions? 5. Can there be a case where two equations represent the same line in such systems? **Tip:** Contradictions in the equations of a system indicate inconsistency, leading to no solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Augmented Matrix
Matrix Consistency

Formulas

Matrix form of a system of linear equations

Theorems

Consistency theorem for systems of equations

Suitable Grade Level

Grades 10-12

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