91Ó°ÊÓ

Linear equations are mathematical expressions that represent straight lines when plotted on a graph. These equations involve variables, usually denoted as \(x\), \(y\), and \(z\), which do not have exponents other than one. Linear equations form the basis of linear algebra and are fundamental in solving problems involving quantities that change at a constant rate.
For a set of linear equations, such as the ones given in the exercise, the goal is to find values for each variable that make all the equations true simultaneously. In three variables, as shown, the system can sometimes be visualized as the intersection of three planes in three-dimensional space. The intersection point (if it exists) represents the solution to the system.

• Always check if the equations are truly linear by inspecting the variables and their exponents.
• Coefficients represent the rate of change in terms of the variables involved.

An augmented matrix is a compact way to represent a system of linear equations. This matrix format includes two parts: the coefficient matrix and a column for the constants from the right side of the equations, separated by a vertical line.
In our exercise, the augmented matrix is written as:
\[ \begin{array}{ccc|c} 1 & 1 & -2 & -3 \ 2 & -1 & 3 & 7 \ 1 & -2 & 5 & 0 \end{array} \]
This matrix displays all necessary information to solve the equations using row operations. It's essential because it transforms the problem into a format that can be easily manipulated algebraically.

• Left side of the matrix (before the line) represents coefficients of variables.
• Right side of the matrix (after the line) represents constant terms.

Row operations are steps we perform on an augmented matrix to simplify it and eventually arrive at the solutions of the system of linear equations. There are three primary types of row operations:

• Swapping two rows.
• Multiplying a row by a nonzero constant.
• Adding or subtracting a multiple of one row to another row.

These operations help in achieving what's known as the row-echelon form or its more refined version, reduced row-echelon form (RREF).
In the exercise, row operations were used to eliminate certain coefficients to simplify the system step by step, aiming for a row-echelon form. This process allows us to see the relationships between the equations more clearly and to determine if a solution exists.

An inconsistent system is when a set of linear equations has no solution. This typically happens when the equations represent parallel planes that do not intersect or, in other terms, contradictions within the system occur.
In our exercise, after applying row operations, we encounter a row in the matrix where all coefficients are zeroes, but the constant is not zero:\[ \begin{array}{ccc|c} 0 & 0 & 0 & -10 \end{array} \]
This indicates that the system is inconsistent because an equation like \(0x + 0y + 0z = -10\) is non-sensical. Consistency in a system is key to finding solutions, and when such inconsistencies appear, it means there's no value of \(x\), \(y\), or \(z\) that can make all the equations true simultaneously.

• Check the matrix thoroughly after row operations for inconsistent rows.
• Understand that an inconsistent system implies no single solution exists that satisfies all equations.