91Ó°ÊÓ

When we come across a system of linear equations, we're looking at a set of equations where each represents a line in a certain-dimensional space. The goal is usually to find the values of the unknowns that satisfy all equations simultaneously.

For example, in the given exercise, the system extracted from the matrix is:
\[ x = -7 \]
\[ y = 3 \]
Each equation represents a line in 2-dimensional space. In this simple system, since each variable is already isolated on one side with a numerical answer on the other, these lines intersect exactly at the points given by the right-hand side of the equations, indicating a single solution.

Representing linear systems in matrix form is incredibly powerful, offering a structured method for solving and analyzing them. In a matrix, rows correspond to equations and columns to variables, creating a visual and computational clarity. The given exercise uses the augmented matrix form:
\[\begin{array}{ll|r}x & y \1 & 0 & -7 \0 & 1 & 3\end{array}\]
The vertical bar separates the coefficients of the variables from the constants. This format is very useful, especially when dealing with more complicated systems, as it sets the stage for techniques like Gaussian elimination and matrix row operations, which help determine solutions efficiently.

Solving linear systems can vary from a straightforward substitution to complex algorithms, depending on the nature of the system. To assess the solvability of a system, we label it as consistent—meaning at least one solution exists, or inconsistent—where no solutions can satisfy all equations simultaneously.

In the exercise, we have a consistent system because we can assign a specific value to each variable without contradiction. The simplicity allows us to read the solution directly: \( x = -7, y = 3 \). The consistency is apparent without needing row operations or advanced methods. However, for more complex systems, one might employ techniques such as Gaussian elimination, graphing, or, for larger systems, matrix inversion or computer algorithms to find the solutions.