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Linear equations are algebraic expressions that represent lines when graphed on a coordinate plane. They have variables raised only to the first power and do not multiply each other. A standard form of a linear equation in three variables is typically written as:

• \( ax + by + cz = d \)

Here, \(a\), \(b\), \(c\), and \(d\) are constants, and \(x\), \(y\), and \(z\) are variables representing unknowns.
In the original exercise, the equations are rearranged into their standard forms to easily identify coefficients:

• \(x + y = 3\)
• \(-y + z = -1\)
• \(x + z = 2\)

Recognizing these equations as lines helps demonstrate how they interact to form a solution: the point where all three intersect.

Row operations are tools used to manipulate rows within a matrix to solve systems of equations. There are three primary row operations you can perform:

• Swapping two rows
• Multiplying a row by a non-zero constant
• Adding or subtracting the multiples of one row to another

In the exercise, row operations are applied to transform the matrix into a simpler form, aiding us in solving it. For example,
- When subtracting Row 1 from Row 3, the equation \(x + z = 2\) changes by eliminating variable \(x\) from Row 3.- Adding Row 2 to Row 3 simplifies it further, resulting in \(0x + 0y + 2z = -2\).
This sequentially eliminates variables which makes it easier to use subsequent steps to find the solution.

Row echelon form is a specific type of matrix form essential for solving systems of linear equations. It simplifies the matrix, making it easier to identify solutions for the variables. A matrix is in row echelon form if:

• All non-zero rows are above rows of zero.
• The first non-zero number from the left (pivot) in a non-zero row is 1.
• Each leading 1 is to the right of the leading 1 in the row above it.

In the exercise, the row echelon form is attained through row operations:- The augmented matrix is adjusted until it becomes: \[ \begin{bmatrix} 1 & 1 & 0 & | & 3 \ 0 & -1 & 1 & | & -1 \ 0 & 0 & 2 & | & -2 \end{bmatrix} \]Here, each leading entry is in a column to the right of the leading entry above it.
This configuration allows for clear back-substitution to solve for \(x\), \(y\), and \(z\). Thus, row echelon form plays a critical role in simplifying the pathway to the solution.