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The substitution method is a common technique used to solve systems of linear equations, where one equation is manipulated to express one variable in terms of the other. Subsequently, this expression is substituted into the other equation. This approach can simplify the system to a single equation with one variable, making it easier to solve.

Let's break down the process using the given exercise. The first step was to isolate one variable, in this case, 'y', from one of the equations. By doing this, the equation was rearranged to obtain an expression for 'y', which read as: \( y = 5 + \frac{5}{3}x \).

The next logical step involved substituting this expression into the other equation, replacing the 'y' term there. This transformed a two-variable system into a single equation. However, when following this process for the given equations, we reached an apparent paradox: the simplification led to the false statement \( 15 = 6 \) which indicates that there is no point that satisfies both equations simultaneously. Thus, the substitution method shows that the system has no solution.

A system of linear equations consists of two or more linear equations with corresponding variables. The solutions to such a system are the points where the equations' graphs intersect, representing a common solution applicable to all individual equations.

In the context of 'systems of linear equations', our focus goes to finding the set of values that satisfy all the equations simultaneously. Graphically, if they are two-variable linear equations, they will be represented as straight lines. The point(s) where these lines intersect are the solutions to the system. When working through the exercise, it was revealed through the inconsistency reached (\(15eq6\)) that the lines do not intersect at all. Thus, we deduce that this particular system of linear equations has no solution, which brings us to the conclusion that the lines are parallel.

Graphing utility verification is a modern tool, pivotal for double-checking the algebraic solutions for systems of linear equations. After performing the substitution method, utilizing a graphing calculator or software can confirm the nature of the system's solutions, be they one, none, or infinitely many.

In the given exercise, while the algebra via the substitution method has led us to a conclusion of no solution, employing a graphing utility allows us to visualize this result. The lines represented by the equations would appear as parallel, which means they never intersect. This graphical representation is an invaluable asset for students as it reinforces the concept that the system does not have a solution, confirming algebraic findings. Such verification serves also as a sanity check, especially useful in complex systems or in cases where students may be prone to making algebraic errors.