https://file.aiquickdraw.com/mathbot/file/17270235703604wpbfv7y.jpg

The problem presents a row-reduced augmented matrix that corresponds to a system of linear equations. The matrix is:

\[
\begin{pmatrix}
1 & 0 & 0 & | & 5 \\
0 & 1 & 0 & | & -1 \\
0 & 0 & 1 & | & 6 \\
\end{pmatrix}
\]

This matrix is in row-reduced echelon form, and from it, we can directly read the solutions to the system. Each row corresponds to one equation:

1. The first row, \( 1 \cdot x + 0 \cdot y + 0 \cdot z = 5 \), simplifies to \( x = 5 \).
2. The second row, \( 0 \cdot x + 1 \cdot y + 0 \cdot z = -1 \), simplifies to \( y = -1 \).
3. The third row, \( 0 \cdot x + 0 \cdot y + 1 \cdot z = 6 \), simplifies to \( z = 6 \).

Thus, the solution to the system is:
\[
x = 5, \, y = -1, \, z = 6.
\]

This corresponds to the first answer option: \([x = 5, y = -1, z = 6]\).

Would you like further details or clarifications on this solution?

Here are five related questions for practice:
1. How can you tell if a system has no solution from the row-reduced form of a matrix?
2. What does it mean if a row in the matrix is entirely zeros?
3. Can a system of equations have an infinite number of solutions? If so, what form does the augmented matrix take?
4. How can you solve a system with more variables than equations?
5. What are the steps to convert a matrix to row-reduced echelon form?

**Tip:** When interpreting an augmented matrix, each row corresponds to one equation in the system. The variables are aligned with each column, making it easy to solve directly when the matrix is in echelon form.

Given that the augmented matrix in row-reduced form below is equivalent to the augmented matrix of a system of linear equations, determine whether the system has a solution and find the solution(s) to the system, if they exist.

Learn how to solve a system of equations by interpreting the augmented matrix in row-reduced echelon form. The matrix represents the system with variables x, y, and z, with a solution of x = 5, y = -1, and z = 6. This problem covers key linear algebra concepts, including solving systems of linear equations using matrix techniques.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation