Solving an Augmented Matrix and Identifying Inconsistency

Math Problem Statement

Determine the solution to the given augmented matrix.

The image you uploaded represents an augmented matrix. It corresponds to a system of linear equations. The matrix is:

1 & 2 & 0 & | & -4 \\ 0 & 0 & 0 & | & -3 \\ 0 & 0 & 1 & | & -9 \end{pmatrix}$$ This system of equations can be written as: $$\begin{aligned} x + 2y &= -4 \\ 0 &= -3 \\ z &= -9 \end{aligned}$$ Now let's analyze the system: 1. The second equation, $$0 = -3$$, is clearly a contradiction and suggests that the system has no solution. Thus, the system is **inconsistent** and has no solution. Would you like me to help explain why this happens, or do you have further questions? Here are some related questions for you: 1. How do you recognize if a system of equations is inconsistent from the augmented matrix? 2. What is the process to solve a system of linear equations using Gaussian elimination? 3. Can a system of equations have infinitely many solutions? What would the augmented matrix look like in that case? 4. How can we represent a consistent and dependent system in augmented matrix form? 5. What happens if the augmented matrix represents a system with more variables than equations? Tip: Always check for rows that represent contradictions, like $$0 = \text{nonzero number}$$, when solving using augmented matrices.

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Linear Algebra
Systems of Linear Equations
Augmented Matrices

Gaussian Elimination
Matrix Row Reduction

Consistency of a System of Equations
Row Echelon Form

Undergraduate (Introductory Linear Algebra) or Advanced High School

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