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. $$ \left[\begin{array}{lll|r} {1} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0} \\ {0} & {0} & {0} & {2} \end{array}\right] $$

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.
$$
\left[\begin{array}{lll|r}
{1} & {0} & {0} & {0} \\
{0} & {1} & {0} & {0} \\
{0} & {0} & {0} & {2}
\end{array}\right]
$$

Key Concepts

Solution of Linear Systems in RREF

Once the matrix is in RREF, the system's solution can be directly read off, with each pivot corresponding to a variable. If the RREF reveals a contradictory equation, it signals that the system is inconsistent.

Consistency and Inconsistency in Linear Systems

A system is consistent if it has at least one solution and inconsistent if there is no solution, which is often revealed in the matrix form when a row results in an impossible equation (for example, all zero coefficients equating to a nonzero constant).

Augmented Matrix Representation

An augmented matrix is a compact way of writing a system of linear equations, with the coefficients of the variables forming the main portion of the matrix and the constants placed in an extra column. This format facilitates systematic procedures, such as Gaussian elimination, to solve the system.

Reduced Row Echelon Form (RREF)

RREF is a canonical form of a matrix achieved via elementary row operations where every leading entry is 1, each leading 1 is the only nonzero entry in its column, and any rows of zeros are at the bottom. This form simplifies the process of solving linear systems by making the relationships between variables explicit.

Interpreting Systems of Linear Equations

A system of linear equations can be translated from its matrix representation by associating each row with an equation where the left part contains the linear combination of variables and the right part represents the constants. This interpretation is the first step in analyzing and solving the system.

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