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Evaluating an algebraic expression is akin to solving a mini mystery. The process involves replacing variables with their actual values and computing the outcome based on the given numbers and operations. In our exercise, we have the expression \(x - 4y\), which is essentially an instruction to subtract four times the value of \(y\) from \(x\).

When we are told that \(x = 2\) and \(y = -3\), the translation is straightforward. We simply plug these values into the expression, resulting in \(2 - 4(-3)\). Here's where understanding the arithmetic arrives on the scene: we multiply first, according to the order of operations. The expression simplifies to \(2 - (-12)\), flipping the subtraction into addition, and voila, we get \(2 + 12\), summing up to 14. This approach is not just about getting the right answer; it’s about ensuring we are fluent in the language of algebra and can interpret expressions accurately.

When we talk about solving equations, we're referring to the search for values that make an equation true. An equation is like a seesaw; it stays balanced when each side is equal. Our task is to maintain this balance by finding the numerical keys—the values of the variables—that lock the seesaw in place.

With the equation \(x - 4y = 14\), and the values \(x = 2\) and \(y = -3\), our job was to test if these values keep our seesaw level. Substituting the given numbers in for \(x\) and \(y\), we performed the arithmetic, stepped through our operations, and confirmed that indeed, these values kept both sides equal at 14. In algebra, this balance is everything; when both sides are equal, the equation is true, and our mission is accomplished.

The solution to our exercise also revisits the basics of arithmetic operations, a foundation without which algebra would crumble. Here, we used two primary operations: multiplication and addition. Our calculation began with multiplying 4 by \(-3\), an operation that involves finding the total of one number taken multiple times. In this case, the product is \(4 \times (-3) = -12\).

Next, we shifted to addition, where we added 2 to our product of -12. However, there's a twist! Since adding a negative is the same as subtracting its positive counterpart, our equation \(2 - (-12)\) turned into \(2 + 12\), which equals 14. This dance of operations, a mix of multiplying and adding (or subtracting), is at the heart of not just this problem but all algebraic manipulation.