What is the distinction between \(x(-y)\) and \(x-y ?\)

Understanding how variables interact through multiplication is essential in algebra. When we talk about the multiplication of variables, we're referring to taking one variable and scaling it by another. For example, when we have the expression \( x(-y) \), it means that we are taking the variable \( x \) and multiplying it by the negative version of the variable \( y \).

Imagine you are stretching or shrinking a piece of string (\textbf{representing \( x \)}) based on another string's length (\textbf{this time, \( -y \) is our scalene factor}). Each variable is like a piece of string, and when you multiply them, you're combining their lengths. In the case of a negative factor like \( -y \), the string's direction is reversed, meaning you're stretching in the opposite way. This is why \( x(-y) \) and \( xy \) are different; the negative sign in front of \( y \) flips the direction of the multiplication.

When variables are subtracted, such as in the expression \( x-y \), we are essentially measuring the difference between them. Think of it as having two towers of blocks, where \( x \) is the height of one tower and \( y \) is the height of the second tower. Subtraction tells you how many blocks the first tower is taller or shorter than the second.

In algebraic terms, subtracting \( y \) from \( x \) means taking away the value of \( y \) from \( x \) . If \( y \) were positive, you'd be removing blocks from tower \( x \) . But since we often deal with negative numbers, as in \( x + (-y) \), it's similar to saying you're taking away a negative amount, which ironically means adding blocks to your tower \( x \) . This concept becomes particularly handy when dealing with algebraic operations that involve negative numbers.

Negative numbers in algebra can sometimes trip up students, but they're just a sign of direction rather than a pitfall. In the context of \( x(-y) \), we see the application of a negative number multiplying a variable. This 'negative' essentially means we're taking the value of \( y \) and moving in the opposite direction from what's considered 'positive'.

If you associate positive numbers with moving forward, then negative numbers would be like taking steps backward. In the operation \( x + (-1 * y) \) , the negative one (\( -1 \) ) multiplies \( y \) to change its direction, making it a subtraction when placed next to \( x \) . It's important to handle negative numbers with care to avoid common mistakes. Always remember that two negatives become a positive, which explains why \( -( -y ) \) turns back into \( y \).

Algebraic operations are the bread and butter of algebra and include addition, subtraction, multiplication, and division. When we apply these operations to variables, we can solve equations and simplify expressions. The order in which these operations are performed is crucial and is guided by the rules of precedence or the BODMAS/BIDMAS rule (Brackets, Orders [powers and roots], Division and Multiplication, Addition and Subtraction).

In our exercise, we are dealing with multiplication and subtraction. Multiplication of variables, like \( x(-y) \), always comes before subtraction in the absence of parentheses that dictate otherwise. This clear structure allows us to systematically approach algebraic expressions and unravel them step by step, ensuring we arrive at the correct solution every time.