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Variable substitution is a foundational technique in algebra that involves replacing variables with their given numerical values. The process is similar to following a simple recipe, where ingredients (variables) are swapped out for the actual food items (numbers) you have on your kitchen shelf.

For example, if we're going through a recipe that calls for 'x' cups of flour and 'y' eggs, and we know 'x' is 3 and 'y' is 2, we'd substitute the 3 for 'x' and 2 for 'y'. In algebraic expressions, this process follows the same logic. If you are given an expression like \( -x + y \), and the specified values, \( x = -5 \) and \( y = -7 \) like in our exercise, you simply replace the 'x' with -5 and 'y' with -7, resulting in \( -(-5) + (-7) \).

Importance of Proper Substitution

Proper substitution is key to ensuring accurate results. If you swap in the wrong numbers or overlook a negative sign, the outcome will be incorrect. This emphasizes the importance of carefully reading the problem and correctly replacing every variable.

Once variables are substituted, simplifying expressions is the next step. It cleans up the equation to make it easier to solve. Think of it as cleaning up after cooking—putting things in order so you can see exactly what you’re left with.

After substitution in our exercise, you should have an expression that looks tangled and messy, \( -(-5) + (-7) \). The simplified form removes any parentheses and combines like terms, reducing the expression to its simplest form. To simplify, we first address the double negative on the 5, which is like saying the opposite of a negative, resulting in a positive. So, \( -(-5) \) simplifies to \( 5 \). Then, we combine it with the remaining term, and it becomes \( 5 - 7 \), a much neater and simpler expression to work with.

Taking Care with Negatives

Misinterpreting negative signs is a common error. It's essential to treat them with care. Overlooking a negative sign during the simplification process can derail the entire problem, leading to a faulty result.

After variable substitution and simplification, it's time to perform arithmetic operations. These are basic math maneuvers—addition, subtraction, multiplication, and division—that we use to compute a final answer. In the context of the simplified equation \( 5 - 7 \), we're faced with a simple subtraction problem.

Subtracting 7 from 5 brings us into negative territory, which can sometimes be a bit counterintuitive. Rather than gaining, we're owing; the answer is \( -2 \). Consider it like having \(5 in your wallet and wanting to buy something that costs \)7. You're short by $2, hence, \( -2 \).

Keeping Operations Straight

It's crucial not to rush and mix up the operations. Subtraction problems are often mistakenly turned into addition, and vice versa, leading to entirely different outcomes. Always double-check the operation required and proceed methodically to ensure precision and avoid introducing errors into your work.