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. $$ \left[\begin{array}{rrr|r} 1 & -3 & -1 & -3 \\ 0 & 1 & -\frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & 2 \end{array}\right] $$

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.
$$
\left[\begin{array}{rrr|r}
1 & -3 & -1 & -3 \\
0 & 1 & -\frac{1}{4} & \frac{1}{2} \\
0 & 0 & 1 & 2
\end{array}\right]
$$

Key Concepts

Back-Substitution

Back-substitution is a method used to solve a system of equations that has been transformed into an upper triangular or row-echelon form. Starting from the last equation, which typically contains only one variable, the value of each variable is determined by substituting the known values back into the previous equations sequentially.

System of Linear Equations

A system of linear equations consists of multiple equations with several variables, where each equation is linear, meaning that each term is either a constant or the product of a constant and a single variable. Understanding systems of linear equations is fundamental in linear algebra, as it provides insights into the intersection of planes, dependency among equations, and the conditions for unique, infinite, or no solutions.

Row-Echelon Form

Row-echelon form is a simplified version of a matrix achieved through elementary row operations, where each nonzero row starts with a leading coefficient (or pivot) that is to the right of the leading coefficient of the row above it, and zeros appear below each pivot. This form makes it easier to identify the structure of the linear system and solve it efficiently.

Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations, where the coefficients of the variables and the constants from the right-hand side are combined into a single matrix. This format is essential for applying systematic methods like Gaussian elimination to simplify and solve the system.

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