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When dealing with linear recurrence relations like the given problem, the characteristic equation is a key concept. It provides a way to identify a pattern or formula that fits all terms in the sequence. To derive the characteristic equation, we start by assuming solutions of the form \( a_n = r^n \), where \( r \) is a constant. This approach works well for linear homogeneous recurrence relations with constant coefficients.

By substituting \( r^n \) into the given relation \( a_n = -5a_{n-1} + 6a_{n-2} \), and then dividing through by \( r^{n-2} \), we obtain a quadratic equation: \( r^2 + 5r - 6 = 0 \). This equation, known as the characteristic equation, serves as a foundational tool to determine the pattern of the sequence.

Initial conditions are essential for finding the particular solution of a recurrence relation. They provide specific values for the first few terms of the sequence, ensuring the solution meets certain criteria. In our example, the initial conditions are \( a_0 = 5 \) and \( a_1 = 19 \).

These initial conditions allow us to calculate the constants in the general solution, tailored specifically to the problem at hand. Without initial conditions, we can only find a general form of the sequence. These starting values create a unique path for how the sequence will evolve. Once we have the initial conditions, they are plugged into the general solution, forming a "system of equations" which we'll solve next.

The general solution is a formula or expression that represents all possible solutions to a recurrence relation before initial conditions are applied. For the recurrence relation \( a_n = -5a_{n-1} + 6a_{n-2} \), and after solving the characteristic equation, we found the roots to be \( r = 1 \) and \( r = -6 \).

These roots enable us to construct the general solution: \( a_n = A(1)^n + B(-6)^n \), where \( A \) and \( B \) are arbitrary constants. The terms \( (1)^n \) and \( (-6)^n \) correspond to the characteristic roots and illustrate the growth or decay pattern of the sequence. The coefficients \( A \) and \( B \) are determined by applying the initial conditions, hence shifting the general form to suit our specific sequence.

A system of equations emerges when we apply initial conditions to the general solution. By substituting \( a_0 = 5 \) and \( a_1 = 19 \) into the general solution, we derive two equations: \( A + B = 5 \) and \( A - 6B = 19 \).

These equations work together to lock in the values of \( A \) and \( B \). Solving this system involves a step-by-step algebraic process: we start by eliminating one variable, typically through substitution or elimination, simplifying the computation.

For instance, here by subtracting the first from the second equation, we simplified the system and found \( B = -2 \). Using this in the first equation, we quickly derived \( A = 7 \). Finally, replacing \( A \) and \( B \) back into the general solution, we completed the process, obtaining the particular solution, which describes our original sequence.