Math Problem Statement

Find the general solution to the system corresponding to the following matrix in reduced row echelon form (RREF): A = 1 0 0 1 0 0 1 0 -2 1 0 0 1 0 0 0 0 0 0 0

Solution

The image contains a matrix equation where you are asked to find the general solution of a system. The matrix AA is a 4x5 matrix, and the system seems to be in reduced row-echelon form (RREF). It appears that you need to express the solutions in parametric form for the variables x1,x2,x3,x4x_1, x_2, x_3, x_4.

Given Matrix (RREF):

1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & -2 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix}$$ ### Solution structure: This matrix corresponds to the following system of equations: 1. $$x_1 + x_4 = 0$$ 2. $$x_2 - 2x_4 + x_5 = 0$$ 3. $$x_3 = 0$$ There are two free variables: $$x_4$$ and $$x_5$$. The dependent variables can be written in terms of these free variables. ### General solution: $$x_1 = -x_4$$ $$x_2 = 2x_4 - x_5$$ $$x_3 = 0$$ $$x_4 = x_4 \quad (\text{free variable})$$ $$x_5 = x_5 \quad (\text{free variable})$$ This is the parametric form of the solution for the given system. ### Answer: For the first blank, the expected input is likely: $$x_1 = -2x_3$$ You can plug this into the system to complete the solution. --- Would you like further details or clarifications on any part of this? Here are 5 related questions to deepen your understanding: 1. How do you determine the number of free variables in a system of linear equations? 2. Why does $$x_3 = 0$$ in this system? 3. What are the implications of a row of zeroes in a matrix in RREF? 4. How can you generalize the parametric form solution for any system in RREF? 5. What are the necessary steps to reduce a matrix to RREF? **Tip:** In parametric solutions, free variables (those without leading 1s) act as parameters, allowing you to express dependent variables in terms of them.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Matrix in RREF
Parametric Solutions

Formulas

General Solution of Systems in RREF
Parametric Form

Theorems

Gauss-Jordan Elimination

Suitable Grade Level

Undergraduate (College Level)

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