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Understanding algebraic operations is crucial when dealing with numbers, including negatives. Let's break it down with our present scenario. We have two negative numbers, say \(a\) and \(b\). The operation we are examining is subtraction, represented by \(a-b\).

Now, visualize negative numbers on a number line: they are to the left of zero. When subtracting one negative number from another, it's akin to removing debt; your amount can actually go up. For example, if you owe \(5 (\(a = -5\)) and you eliminate a debt of \)3 (\(b = -3\)), you are essentially doing \( -5 - (-3) \) which simplifies to \( -5 + 3 \) – a less negative outcome of $2 in debt. In algebra, subtracting a negative is the same as adding its positive counterpart; \( -(-y) \) becomes \( +y \).

This can lead to three possibilities: if the magnitude of the first number (\(x\)) is larger, the result is negative; if they're equal, it's zero; and if the magnitude of the first is smaller, the result is positive. Subtraction among negative numbers isn't always intuitive, but remembering that two negatives make a positive can help you navigate through these problems.

Real numbers include the complete set of rational and irrational numbers, which can be positive, negative, or zero. They fall within a continuum on the number line, stretching indefinitely in both directions. This set includes the integers, fractions, and decimals we encounter daily. In our exercise, \(a\) and \(b\) are negative real numbers, which means they are less than zero but still a part of the broad family of real numbers.

Why does it matter? Because the properties of real numbers help us solve algebraic problems. For instance, subtraction is commutative over real numbers, allowing for the operation \(a - b\) to be understood as \(a + (-b)\), which is precisely what happens when you subtract a negative number: you are adding its positive counterpart. Another key point is knowing that the result of operations between any two real numbers is always another real number. There are no mysterious outcomes in standard algebra that can't be pinpointed on the number line of real numbers.

Variables are symbols used to represent unknown values and are fundamental in algebra. They can take on various values from a specified set, often real numbers. In the given exercise, we are dealing with variables \(a\) and \(b\), which are pretenders for negative numbers. By assigning variables, \(a = -x\) and \(b = -y\), where \(x\) and \(y\) are positive, we've introduced an abstraction layer that simplifies our problem-solving process.

Think of variables as placeholders or empty containers waiting to be filled with values that meet the problem's conditions. In algebraic operations, we manipulate these variables using set rules that govern arithmetic while preserving their inherent nature. When we operate with variables, we're using a level of generality that allows us to solve a multitude of problems at once. The subtlety and power of algebra come alive when we use variables to capture and express general truths about numbers, such as the outcome of subtracting two negative numbers.