Math Problem Statement
The linear system corresponding to the augmented matrix:
[1 3 | 5
2 6 | 11]
a) has infinitely many solutions
b) has a unique solution
c) has two solutions
d) has no solution
e) None of the above
Solution
The given problem involves determining the solution set for a system of linear equations based on the augmented matrix:
1 & 3 & | & 5 \\ 2 & 6 & | & 11 \end{pmatrix}$$ ### Step 1: Write the system of linear equations The augmented matrix represents the following system of equations: $$1x + 3y = 5 \quad \text{(Equation 1)}$$ $$2x + 6y = 11 \quad \text{(Equation 2)}$$ ### Step 2: Solve the system using elimination We can use the elimination method by multiplying Equation 1 by 2 and subtracting it from Equation 2 to eliminate $$x$$: $$2(x + 3y) = 2(5) \quad \Rightarrow \quad 2x + 6y = 10$$ Now, subtract this from Equation 2: $$(2x + 6y) - (2x + 6y) = 11 - 10$$ $$0 = 1$$ This is a contradiction, meaning the system is inconsistent and has no solution. ### Conclusion: The correct answer is **d) has no solution**. Would you like a more detailed explanation or have any questions? Here are 5 related questions for further exploration: 1. How do you recognize an inconsistent system from its augmented matrix? 2. What is the geometric interpretation of a system that has no solution? 3. How would the solution change if the augmented matrix were different? 4. Can a system with more equations than variables still have no solution? 5. How is row-reduction related to solving systems of equations? **Tip:** Always check for inconsistencies when using elimination, especially if you encounter equations like $$0 = 1$$, which signal no solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Representation
Gaussian Elimination
Formulas
Augmented Matrix Form
Row Reduction Method
Theorems
Inconsistent System Theorem
Row Reduction Inconsistency
Suitable Grade Level
Grades 10-12