Solve each system of equations using marrices (row...

Solve each system of equations using marrices (row operations). If the system has no solution, say that it is inconsistent.
$$
\left\{\begin{aligned}
2 x+y &=-4 \\
-2 y+4 z &=0 \\
3 x-2 z &=-11
\end{aligned}\right.
$$

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Key Concepts

Augmented Matrix

An augmented matrix is a concise representation of a system of linear equations, where each row corresponds to an equation and each column corresponds to the coefficients of the variables, with an additional column for the constants. This format is essential for applying systematic techniques like row operations to solve the system.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations by performing elementary row operations (such as row swapping, scaling, and adding a multiple of one row to another) on the augmented matrix. This process transforms the matrix into a simpler form, often row-echelon form, from which the solutions can be easily obtained.

Consistency and Inconsistency

The concepts of consistency and inconsistency in a system of equations refer to whether the system has at least one solution or none at all. A system is consistent if it has one or more solutions and inconsistent if it leads to a contradiction during the process of row reduction (for example, a row that reads 0 = nonzero).