Write the system of linear equations represented by the...

Write the system of linear equations represented by the augmented matrix. Use $x, y,$ and $z,$ or, if necessary, $w, x, y,$ and $z,$ for the variables.
$$\left[\begin{array}{rrrr|r}
4 & 1 & 5 & 1 & 6 \\
1 & -1 & 0 & -1 & 8 \\
3 & 0 & 0 & 7 & 4 \\
0 & 0 & 11 & 5 & 3
\end{array}\right]$$

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Key Concepts

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations where the columns correspond to the coefficients of the variables and an extra column represents the constant terms of each equation. This format allows for systematic techniques, such as row reduction, to solve the system.

System of Linear Equations

A system of linear equations is a collection of equations where each equation is linear in the unknown variables. The objective is to find all possible values for these variables that satisfy all the equations simultaneously.

Coefficient Matrix

The coefficient matrix is the part of the augmented matrix that contains only the coefficients of the variables in the system. It encapsulates the relationships between the individual variables without including the constant terms.

Variables

Variables represent the unknown quantities in a system of equations and are typically denoted by letters like x, y, and z, with additional letters (such as w) used if there are more unknowns than the standard three. They are essential for formulating and solving the equations.

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