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.
$$\begin{aligned}
x-2 y+z &=4 \\
2 x+2 y-2 z &=-1 \\
x+4 y+z &=7
\end{aligned}$$

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. Solving such a system means finding the values of the variables that satisfy all the equations simultaneously. This concept is foundational in algebra and has applications across various fields including engineering, economics, and sciences.

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations. It includes the coefficients of the variables as well as the constants from the right-hand sides of the equations. This representation simplifies the manipulation and application of systematic procedures such as row operations to solve the system.

Elementary Row Operations

Elementary row operations are the basic manipulations that can be performed on the rows of a matrix. These operations include swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. They are fundamental for transforming matrices into simpler forms without changing the solution set of their corresponding system of equations.

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations. It involves using elementary row operations to transform the augmented matrix into an upper triangular form, from which the solutions can be efficiently derived. This method is one of the most common algorithms taught for solving such systems.

Back Substitution

Back substitution is the process used after achieving an upper triangular form through Gaussian elimination. Once the matrix is in triangular form, the value of one variable (typically the one in the last row) is found first, and then this value is substituted back into the previous equations to solve for the remaining variables, ultimately leading to the solution of the system.

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