Math Problem Statement
Analyze the augmented matrix and determine the solution to the system of equations represented by it.
Solution
The augmented matrix provided in the image represents a system of linear equations. To determine the solution, we analyze the rows of the matrix:
The augmented matrix is:
1 & 2 & 0 & | & 3 \\ 0 & 0 & 1 & | & -2 \\ 0 & 0 & 0 & | & 5 \end{bmatrix}$$ ### Row Analysis: 1. **Row 1**: $$1x + 2y + 0z = 3$$ 2. **Row 2**: $$0x + 0y + 1z = -2$$, which implies $$z = -2$$. 3. **Row 3**: $$0x + 0y + 0z = 5$$, which simplifies to $$0 = 5$$. This is a contradiction because $$0 \neq 5$$. Therefore, the system is inconsistent. ### Conclusion: The system has **no solution**, as indicated by the inconsistency in Row 3. The correct choice is: - **"The system has no solution."** --- Would you like further clarification or a breakdown of why contradictions lead to no solutions? Here are five additional questions for exploration: 1. How can you identify an inconsistent system in a different matrix? 2. What happens when a system has a row of all zeros with a zero in the augmented column? 3. How does Gaussian elimination confirm the inconsistency of this matrix? 4. What are the different classifications for solutions of a system of equations? 5. Can you provide a real-world example of when a system might have no solution? **Tip**: Always check the augmented part of the matrix (to the right of the bar) for contradictions when analyzing rows of zeros!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Representation
Formulas
Gaussian Elimination
Row Reduction
Theorems
Inconsistency in Systems of Equations
Suitable Grade Level
Undergraduate (Introductory Linear Algebra)