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Solving linear equations often requires you to isolate the variable you're solving for. This means moving all instances of that variable to one side of the equation, often using addition or subtraction.

For instance, take the equation from the exercise, where you have to solve for the variable \(y\) in \(Ax + By = Cy + D\). To 'isolate' \(y\), you combine like terms that contain \(y\) by subtracting \(Cy\) from both sides, which gives you \(Ax + By - Cy = D\). Then, you simplify the equation to \(Ax + (B-C)y = D\), effectively isolating \(y\) terms on one side of the equation. This strategic manipulation is the cornerstone of solving linear equations and is essential to finding a solution.

In actions where both sides of the equation are affected equally (like adding, subtracting, multiplying, or dividing both sides by the same number), you ensure that the equality holds true. Remember, the ultimate goal is to have the variable alone on one side, ready to be solved.

Once variables are isolated, solving linear equations is about finding the solution that satisfies the equation. Continuing the previous example, after isolating \(y\), you obtain \(Ax + (B-C)y = D\), which is one step away from finding \(y\).

To have \(y\) completely isolated—meaning on one side of the equation by itself—you divide each term of the equation by \((B-C)\). This gives you the solution: \[ y = \frac{{D - Ax}}{{B - C}} \].

In this found form, \(y\) is expressed in terms of the other variables and constants in the equation, namely \(A\), \(B\), \(C\), \(D\), and \(x\). This relation is the 'solution' to the original equation, providing a way to calculate \(y\) for any given values of the other variables, assuming they satisfy the required conditions—such as not making any denominators equal to zero.

A crucial consideration when solving linear equations is the prevention of a division by zero error. This mistake occurs when an equation is divided by a variable or expression that can be equal to zero.

Consider the solution \[ y = \frac{{D - Ax}}{{B - C}} \]. The denominator \((B-C)\) must not be equal to zero, because division by zero is undefined in mathematics. Not only does it potentially lead to an incorrect solution, but it also can cause computational errors in calculators and computer programs.

Therefore, as a part of solving any equation, it's vital to check the terms you're dividing by and explicitly state that these cannot be zero. In educational contexts, noting these constraints highlights a deeper understanding of the mathematics involved and the potential for errors in the process, ensuring a valid solution in any case where the equation is applied.