Math Problem Statement

Hvor mange løsninger har ligningssystemet beskrevet ved den udvidede matrix: \[ \begin{bmatrix} 1 & 1 & 3 & 1 \\ 0 & \frac{1}{2} & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

Solution

This is a problem asking for the number of solutions to a system of linear equations based on the given augmented matrix:

1 & 1 & 3 & 1 \\ 0 & \frac{1}{2} & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ ### Steps to Determine the Number of Solutions: 1. **Analyze the Matrix**: The third row is completely zero, meaning it's a dependent equation, which does not provide any new information. The system effectively consists of two equations. 2. **Row Reduction**: The matrix is already close to row-echelon form (REF). We simplify the second row by multiplying it by 2 to get rid of the fraction: $$\begin{bmatrix} 1 & 1 & 3 & 1 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ 3. **Interpret the System**: - The first row corresponds to the equation: $$x_1 + x_2 + 3x_3 = 1$$ - The second row corresponds to the equation: $$x_2 + 2x_3 = 2$$ - The third row is all zeros, indicating it is a redundant equation. 4. **Determine the Number of Variables and Equations**: There are three unknowns $$x_1, x_2, x_3$$ and two independent equations. This leads to a **free variable** ($$x_3$$). 5. **Conclusion**: Since there is one free variable ($$x_3$$), the system has **infinitely many solutions**. The solution will depend on the value assigned to $$x_3$$. ### Final Answer: The system has **infinitely many solutions**. --- Would you like further details or have any other questions? Here are 5 related questions to deepen understanding: 1. How does the concept of free variables affect the number of solutions in a system of equations? 2. Can you explain the difference between a unique solution, no solution, and infinitely many solutions? 3. How would this problem change if the third row were non-zero? 4. What are the criteria for a matrix to have no solutions based on its augmented form? 5. How can you use the rank of a matrix to determine the number of solutions? **Tip:** In general, a system with fewer independent equations than unknowns (i.e., more variables than rows of non-zero information) will have infinitely many solutions if it is consistent.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Row Reduction

Formulas

Matrix Row-Echelon Form

Theorems

Free Variable and Consistency of Linear Systems

Suitable Grade Level

Grades 10-12

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