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Linear Algebra is a branch of mathematics that deals with vectors, vector spaces (also called linear spaces), and linear transformations. It provides a framework for analyzing linear equations and their solutions. In the context of solving systems of equations, Linear Algebra involves finding values for variables that satisfy all equations simultaneously. This is crucial in various fields, like engineering, physics, computer science, and economics.
In a linear equation system, each equation corresponds to a straight line when graphed. A solution to the system is a point where all these lines intersect, provided that such a point exists. If the lines are parallel, there might be no solution, or if they overlap entirely, there could be infinitely many solutions.
This fascinating area helps us understand and work with numerous practical problems, such as optimization problems, modeling natural phenomena, and even machine learning tasks.

Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. They provide a concise way to write and manipulate systems of linear equations. The matrix representation of equations is powerful for solving these systems using technology.
For example, in our exercise, the system of equations is written in matrix form as a multiplication of a coefficient matrix, a variable matrix, and a result matrix. This can be solved using matrix operations like Gaussian elimination or by computing the inverse of a matrix (if it exists) using software tools.
By entering the matrix in technology like MATLAB or a graphing calculator, one can solve for the unknowns efficiently, checking various solutions in a much easier and faster way than solving manually.

Fractional Solutions are often used in mathematics to express answers more precisely. They are particularly useful because they provide exact values rather than decimal approximations, which can sometimes distort the true value due to rounding errors.
In the context of solving the system of equations we're discussing, the solution was represented as fractions like \(\frac{99}{65}\) instead of decimals. This was done to ensure precision, especially when verifying solutions. Representing solutions in fractional form is essential when accuracy is paramount, such as in scientific computations or financial calculations.
It's important to practice converting decimal solutions to fractions and vice versa, leveraging tools that automate such transformations and understanding their applications in solving real-world problems.

Verification of Solutions is a critical step in solving systems of equations. It ensures that the answers derived are correct and satisfy the original equations. This step involves substituting the obtained values back into the initial set of equations.
For instance, in the given exercise, we substituted the fraction-based solution back into the equations to see if they satisfy the equations. Each equation in the original system was checked to ensure the left and right sides balance out.
Using this verification process is vital in learning, as errors in initial solutions can be corrected early on, and it provides confidence that the mathematical process was executed correctly. This step is a necessary habit for both students and professionals dealing with mathematical computations.