https://file.aiquickdraw.com/mathbot/file/17332526314871hlsfswf.jpg

 sympy import Matrix

# Define the augmented matrix from the given equations
matrix = Matrix([
    [5, 3, 5, 12, 10],
    [2, 2, 3, 5, 4],
    [1, 7, 9, 4, 2]
])

# Perform Gaussian elimination
reduced_matrix = matrix.rref()

reduced_matrixUsing Gaussian elimination, the system of equations reduces to the following:

\[
\begin{aligned}
x_1 + \frac{1}{4}x_3 + \frac{9}{4}x_4 &= 2, \\
x_2 + \frac{5}{4}x_3 + \frac{1}{4}x_4 &= 0.
\end{aligned}
\]

The third row indicates there are no further constraints, implying the system has infinitely many solutions. The solutions can be expressed parametrically as:

\[
\begin{aligned}
x_1 &= 2 - \frac{1}{4}x_3 - \frac{9}{4}x_4, \\
x_2 &= -\frac{5}{4}x_3 - \frac{1}{4}x_4, \\
x_3 &= x_3, \\
x_4 &= x_4,
\end{aligned}
\]

where \(x_3\) and \(x_4\) are free variables.

Let me know if you need further clarification or want to explore a particular solution. Here are some follow-up questions for better understanding:

1. Would you like to set specific values for \(x_3\) and \(x_4\) to obtain particular solutions?
2. Do you need a step-by-step explanation of the Gaussian elimination process?
3. Would you like to visualize the parametric solution set geometrically?
4. Do you want to explore how this method applies to other similar systems of equations?
5. Should I verify these results with another numerical method?

**Tip:** Always verify if the augmented matrix's rank matches the coefficient matrix's rank to check for system consistency!

Discover how to solve a system of linear equations using Gaussian elimination. This example involves three equations with four variables and demonstrates the reduction of the system to parametric solutions. Explore key concepts in linear algebra suitable for advanced high school or undergraduate students.

Math Problem Statement

Solution

Define the augmented matrix from the given equations

Perform Gaussian elimination

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation