Each augmented matrix is in row-echelon form. Assume that...

Each augmented matrix is in row-echelon form. Assume that the variables are $x, y,$ and $z$ and use back-substitution to obtain the solution of the associated system of linear equations.
$$
\left[\begin{array}{ll|l}
1 & \frac{3}{2} & 2 \\
0 & 1 & 1
\end{array}\right]
$$

Key Concepts

Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations where the coefficients of the variables are written in a matrix alongside an additional column that contains the constants from the equations. This format is especially useful when applying systematic methods such as Gaussian elimination to find the solutions of the system.

Row-Echelon Form

A matrix is in row-echelon form when each nonzero row starts with a leading coefficient (also called a pivot) that is positioned to the right of the leading coefficient in the row above it, and any rows consisting entirely of zeros are at the bottom. This form simplifies the process of solving the system since it often allows direct application of back-substitution.

Back-Substitution

Back-substitution is a method used to solve a system of linear equations once the matrix has been transformed into a triangular or row-echelon form. Starting from the last equation, where one variable is isolated, the known values are substituted back into the preceding equations to sequentially solve for the remaining variables.

Free Variables

Free variables occur in a system of linear equations when there are fewer pivot positions than variables, meaning that not every variable is leading. These variables are not uniquely determined by the system and can take on any value, often leading to infinitely many solutions which are expressed in parametric form.

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