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Understanding how to solve algebraic equations is a fundamental skill in mathematics. An equation is like a balancing scale; whatever you do to one side, you must do to the other to keep it balanced. Specifically, when you encounter an equation such as \( 4w = 108 \), your goal is to find the value of the unknown variable, in this case, \( w \). To do this, you can employ various properties of equality, one of which is the division property.

This property allows you to divide both sides of the equation by the same non-zero number, thus maintaining the balance. Imagine you have 4 bags containing the same number of marbles that total 108 marbles. To find out the marbles per bag (the value of \( w \)), you'd divide the total number of marbles by the number of bags. Similarly, in our equation, by dividing both sides by 4, you isolate \( w \) and find it equals 27. This simplifies the original complex expression into an understandable answer: \( w = 27 \).

After solving an algebraic equation, it's crucial to verify that your solution is indeed correct. This step ensures that no errors were made during the solving process. Verifying an equation is akin to retracing your steps to ensure no errors were made along the way. For the equation \( 4w = 108 \), after you have calculated that \( w = 27 \), you substitute 27 back into the original equation in place of \( w \).

If the two sides are equal after substitution, your solution holds up—much like checking if a key fits a lock. Performing this check, you multiply 4 by 27 to get 108, which matches the original right-hand side of the equation. Thus the solution \( w = 27 \) is confirmed as correct. This is an essential part of solving any algebraic problem, giving you confidence in your answer.

Isolating the variable is a crucial step in solving algebraic equations. It involves manipulating the equation to get the variable on one side by itself. The objective is to 'isolate' the variable so that you end up with an equation of the form \( variable = value \). For example, with the equation \( 4w = 108 \), the variable \( w \) needs to be isolated.

To achieve this, division is often required, especially when the variable has been multiplied by a factor—like the 4 in our exercise. By dividing both sides of the equation by this factor, you effectively cancel out the multiplication, leaving the variable 'alone' on one side of the equation. This process is a critical strategy in algebra that allows you to solve for unknown quantities and is widely applicable across various types of mathematical problem-solving.