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

Write the system of linear equations represented by the augmented matrix. (Use the variables $x, y, z,$ and $w,$ if applicable.) $$\left[\begin{array}{rrrrrr} 6 & 2 & -1 & -5 & 5 & -25 \\ -1 & 0 & 7 & 3 & \vdots & 7 \\ 4 & -1 & -10 & 6 & \vdots & 23 \\ 0 & 8 & 1 & -11 & \vdots & -21 \end{array}\right]$$

Write the system of linear equations represented by the augmented matrix. (Use the variables $x, y, z,$ and $w,$ if applicable.)
$$\left[\begin{array}{rrrrrr}
6 & 2 & -1 & -5 & 5 & -25 \\
-1 & 0 & 7 & 3 & \vdots & 7 \\
4 & -1 & -10 & 6 & \vdots & 23 \\
0 & 8 & 1 & -11 & \vdots & -21
\end{array}\right]$$

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Key Concepts

System of Linear Equations

A collection of equations involving the same set of variables where each equation is linear. The solution to the system is a set of variable values that simultaneously satisfy every equation in the system.

Augmented Matrix

A compact way to represent a system of linear equations by combining the coefficients of the variables and the constants from the right-hand side into one matrix. Each row corresponds to an equation, and rightmost column represents the constants.

Representation of Equations by Matrix Rows

In an augmented matrix, each individual row symbolizes a single linear equation. The entries in a row before the vertical divider are the coefficients of the variables, and the entry after the divider is the constant term of the equation.

Variable Notation in Linear Systems

In the context of systems of equations, variables such as x, y, z, and w represent the unknowns. Their consistent use aids in setting up the matrix and interpreting the solution to the system.

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