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. $$ \left[\begin{array}{lll|l} 1 & 1 & 4 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & 0 & 1 & \frac{1}{2} \end{array}\right] $$

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.
$$
\left[\begin{array}{lll|l}
1 & 1 & 4 & 6 \\
0 & 1 & 2 & 4 \\
0 & 0 & 1 & \frac{1}{2}
\end{array}\right]
$$

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Key Concepts

Back-Substitution

Back-substitution is a method used to solve a system of equations in row-echelon form. Starting from the last nonzero row and its corresponding variable, one substitutes the found value into the preceding equations to progressively solve for the remaining variables.

Row-Echelon Form

Row-echelon form is a type of matrix form where each successive row has more leading zeros than the previous row. This triangular structure simplifies the process of solving a system of equations because it allows for a systematic approach, often culminating in back-substitution.

System of Linear Equations

A system of linear equations consists of multiple equations where each equation is linear, meaning that each variable is only to the first power and appears as a term in a sum, possibly multiplied by a constant. Such systems can be solved to find values for the variables that satisfy all equations simultaneously.

Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations that includes both the coefficients of the variables and the constants from the right-hand side of the equations. This matrix form enables the use of matrix operations to simplify the system, leading towards a solution.

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