Use row operations on an augmented matrix to solve each...

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate.
$$\begin{aligned} x+3 y-6 z &=7 \\ 2 x-y+2 z &=0 \\ x+y+2 z &=-1 \end{aligned}$$

Key Concepts

Gaussian Elimination

Gaussian elimination is a step-by-step procedure that uses elementary row operations to transform an augmented matrix into a simpler form, typically row echelon form or reduced row echelon form. This method allows one to solve for the variables sequentially by back-substitution after the matrix has been simplified.

Rounding

Rounding refers to the process of reducing the number of decimal places in numerical answers, which is often necessary in practical applications. When solving systems of linear equations, rounding to a specified precision, such as the nearest thousandth, ensures that the results are presented in a clear and manageable form while acknowledging the limits of numerical accuracy.

Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations where the coefficients and constants are organized into a matrix. This method simplifies the use of algebraic techniques, such as row operations, to solve the equations.

Elementary Row Operations

Elementary row operations are the fundamental manipulations that can be performed on rows of a matrix without changing the solution set of the corresponding system of linear equations. These operations include swapping two rows, multiplying a row by a nonzero scalar, and adding or subtracting a multiple of one row to another, which are essential in matrix simplification techniques.

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