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The substitution method is a powerful technique in solving systems of differential equations. It involves replacing one variable with another equation or expression, which simplifies the system to a single variable scenario. In our exercise, the substitution method comes into play right from the start. After isolating the function x(t) in the first equation, we substitute it into the second, effectively turning the original system into a single differential equation solely in terms of y(t).

This method can tremendously reduce the complexity of a system, making it more manageable. It is essential, however, to track the substitutions made accurately to back-substitute later and retrieve the solution for the initial variables.

A differential equation is a mathematical equation that relates some function with its derivatives. In real-world applications, these equations describe various phenomena like growth, decay, oscillation, and much more. The system we are dealing with in our exercise consists of two first-order differential equations. They are termed 'first-order' because they involve the first derivatives of the variables x(t) and y(t).

Solving differential equations can be straightforward or very complex, depending on the form and order of the function and its derivatives. Thus, understanding the nature of the differential equation at hand is crucial before applying any method to solve it.

When solving non-homogeneous linear differential equations like the one we encounter after substitution in our problem, one method that often comes to use is the method of undetermined coefficients. This technique is handy when dealing with polynomial, exponential, or sinusoidal non-homogeneous terms.

The idea behind this method is to assume a solution of a specific form and then determine the coefficients by plugging the assumed solution into the differential equation. It's a trial and error method that requires a good guess informed by the non-homogeneous part of the equation to work effectively. Proper application of this technique can greatly simplify finding the particular solution to the differential equation.

The Laplace transform is a widely used integral transform in mathematics, especially for analyzing linear time-invariant systems. By transforming a given function of time f(t) into a function of a complex variable s, the Laplace transform converts differential equations into algebraic equations, which are generally easier to solve.

In our textbook problem, should the undetermined coefficients method be unsuitable, the Laplace transform could be employed to solve the differential equation resulting from the substitution. After applying the Laplace transform, solving the algebraic equations for the transformed functions, and finding the inverse Laplace transform, you would obtain the original functions x(t) and y(t). This method is particularly effective for handling initial value problems and non-homogeneous differential equations.