https://file.aiquickdraw.com/mathbot/file/17293688699949lv30vt7.jpg

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

\[
\begin{bmatrix}
1 & 1 & 3 & 1 \\
0 & \frac{1}{2} & 1 & 1 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]

### 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.

Hvor mange løsninger har ligningssystemet beskrevet ved den udvidede matrix:

\[
\begin{bmatrix}
1 & 1 & 3 & 1 \\
0 & \frac{1}{2} & 1 & 1 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]


Determine the number of solutions to a system of linear equations based on the given augmented matrix. This problem involves concepts in linear algebra, such as matrix row reduction and free variables. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation