Reduced Row Echelon Form of a System of Linear Equations

Math Problem Statement

Put the augmented matrix of the following system of linear equations into reduced row echelon form: y + z = 3 x + 2y - z = 2 2x + 4y + z = 7.

It seems the text extracted from the image is incomplete. Based on your image, the system of equations is:

y + z &= 3 \\ x + 2y - z &= 2 \\ 2x + 4y + z &= 7 \end{aligned}$$ Let's form the augmented matrix from this system of equations: $$\left[\begin{array}{ccc|c} 0 & 1 & 1 & 3 \\ 1 & 2 & -1 & 2 \\ 2 & 4 & 1 & 7 \end{array}\right]$$ Now, I'll reduce this matrix to its row echelon form (REF) and further to its reduced row echelon form (RREF).The reduced row echelon form (RREF) of the augmented matrix is: $$\left[\begin{array}{ccc|c} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 \end{array}\right]$$ This corresponds to the solution $$x = -1$$, $$y = 2$$, and $$z = 1$$ for the system of equations. Would you like more details or have any further questions? Here are 5 related questions to deepen your understanding: 1. How is an augmented matrix different from a coefficient matrix? 2. What are the steps involved in transforming a matrix to row echelon form (REF)? 3. How does the pivot position in a matrix determine linear independence? 4. Can you solve this system using substitution or elimination methods? How would that compare? 5. What are some applications of reduced row echelon form in solving real-world problems? **Tip:** When converting a matrix to RREF, always ensure the leading coefficient of each row is 1 and that each leading coefficient column contains zeros above and below it.

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Linear Algebra
Matrices
Row Echelon Form
Systems of Equations

Matrix Augmentation
Row Operations

Row Echelon Form Theorem
Reduced Row Echelon Form Theorem

Grades 10-12 or College-level Linear Algebra

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