The reduced row echelon form of a system of linear equations...

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y ;$ or $x_{1}, x_{2}, x_{3}, x_{4}$ as variables. Determine whether the consistem or inconsistent. If it is consistent, give the solution. $$ \left[\begin{array}{ccc|c} {1} & {0} & {0} & {1} & {-2} \\ {0} & {1} & {0} & {2} & {0} \\ {0} & {0} & {1} & {-1} & {0} \\ {0} & {0} & {0} & {0} \end{array}\right] $$

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y ;$ or $x_{1}, x_{2}, x_{3}, x_{4}$ as variables. Determine whether the consistem or inconsistent. If it is consistent, give the solution.
$$
\left[\begin{array}{ccc|c}
{1} & {0} & {0} & {1} & {-2} \\
{0} & {1} & {0} & {2} & {0} \\
{0} & {0} & {1} & {-1} & {0} \\
{0} & {0} & {0} & {0}
\end{array}\right]
$$

Key Concepts

Extracting Solutions from RREF

Once the augmented matrix is in RREF, each row corresponds directly to an equation where the leading variables are expressed in terms of the free variables if any. This facilitates the direct reading of the solution: if each variable has a pivot, the solution is unique; if one or more variables are free, the general solution is given as parameterized in terms of these free variables.

Free Variables in Systems of Equations

Free variables occur when the number of variables exceeds the number of pivot positions in the RREF of the augmented matrix. These variables are not leading in any row and can take arbitrary values, which usually leads to an infinite family of solutions. The solution is then expressed in terms of these free parameters.

Consistency of Linear Systems

A system of linear equations is consistent if it has at least one solution, and inconsistent if it has no solutions. When a system is in RREF, inconsistency can be identified by the presence of a row with all zeros in the coefficient part but a nonzero entry in the constant part, which would correspond to an impossibility like 0 = b (with b nonzero).

Reduced Row Echelon Form (RREF)

The reduced row echelon form of a matrix is a simplified version achieved through elementary row operations. In RREF, each nonzero row begins with a leading 1 (called a pivot), and all entries in the pivot's column are zero except for the pivot itself. This form makes it straightforward to identify the solutions of the corresponding system of equations.

Augmented Matrix Representation

An augmented matrix is a compact way to represent a system of linear equations. The columns of the matrix correspond to the coefficients of the variables in the equations, and the rightmost column represents the constants from each equation. This format simplifies operations such as row reduction and helps in systematic solution of the system.

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