https://file.aiquickdraw.com/mathbot/file/17255051273707ywseg2e.jpg

The image contains a matrix system and its transformation into reduced row echelon form (RREF).

The system is initially given as:
\[
\left\{
\begin{aligned}
    x + y &= 3 \\
    2x + 2y &= 6
\end{aligned}
\right.
\]

This system is represented in matrix form as:
\[
\left[ \begin{array}{cc|c}
    1 & 1 & 3 \\
    2 & 2 & 6
\end{array} \right]
\]

After applying row operations to reduce this augmented matrix to its reduced row echelon form (RREF), we obtain:
\[
\left[ \begin{array}{cc|c}
    1 & 1 & 3 \\
    0 & 0 & 0
\end{array} \right]
\]

### Interpretation:
- The second row of zeros indicates that the system has infinitely many solutions because the second equation does not provide any new information.
- The system reduces to:
  \[
  x + y = 3
  \]
  which has infinitely many solutions of the form \(x = 3 - y\), where \(y\) can be any real number.

Would you like further details on solving systems of linear equations or matrix reduction?

Here are 5 questions to expand:
1. How do row operations work to reduce matrices to RREF?
2. What are the possible outcomes (solution types) for systems of linear equations?
3. How can this system be solved using substitution or elimination methods?
4. What conditions lead to a system having infinitely many solutions versus a unique solution?
5. Can graphing these equations give insight into their solution?

**Tip**: When you encounter a row of zeros in an augmented matrix, it often signals that the system is dependent, leading to infinitely many solutions or no solution, depending on the corresponding constants.

Solve the system of linear equations using matrix reduction to find the reduced row echelon form (RREF).

Learn how to solve a system of linear equations using matrix reduction to find its reduced row echelon form (RREF). This guide explains key concepts in linear algebra, including matrix operations, row operations, and theorems like Rouché–Capelli. 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