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.
$$
\left\{\begin{aligned}
2 x+y-z &=4 \\
-x+y+3 z &=1
\end{aligned}\right.
$$

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Key Concepts

Row Operations

Row operations involve modifying the rows of a matrix through actions such as swapping rows, multiplying a row by a non-zero constant, or adding a multiple of one row to another. These operations are fundamental in transforming the matrix into forms like echelon or reduced row-echelon form, making the system easier to solve.

Free Variables/Parameterization

In systems with fewer independent equations than variables, some variables cannot be uniquely determined and are classified as free variables. The solutions are then expressed in terms of these free parameters, resulting in infinitely many solutions represented by a parametric form.

System of Linear Equations

A system of linear equations is a set of equations where each equation is linear in the variables. These systems can be analyzed and solved to find the values for the variables that satisfy all equations simultaneously.

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations, where the matrix contains both the coefficients of the variables and the constants from the right-hand side of the equations. This format simplifies the application of row operations in solving the equations.

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