Solve System of Linear Equations Using Matrix Reduction to RREF

Math Problem Statement

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

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

The system is initially given as:

\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.

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Linear Algebra
Matrix Operations
Systems of Linear Equations

Row Operations
Reduced Row Echelon Form (RREF)

Rouché–Capelli Theorem
Existence and Uniqueness Theorem for Linear Systems

Grades 10-12

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