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The distributive property is a fundamental algebraic concept used to simplify expressions. It allows you to multiply a single term across terms inside a parenthesis. Think of it as distributing goodies from a gift bag to several friends! For instance, if you have \(-2(xy + 7)\), use the distributive property to multiply \(-2\) with each term inside the parenthesis. Therefore,

• \(-2 \cdot xy = -2xy\)
• \(-2 \cdot 7 = -14\)

Similarly, apply the negative sign \(-1\) across terms inside another parenthesis \((yx - 1)\). Remember that distributing a negative sign changes the sign of each term:

• \(-1 \cdot yx = -yx\)
• \(-1 \cdot -1 = +1\)

After distributing, you'll have the expression: \(-9 - 2xy - 14 - yx + 1\).
Your ability to manage both numbers and variables with this property will make simplification much easier!

Combining like terms is another essential simplification step. Like terms are terms whose variables are the same, raised to the same power. For example, \(-2xy\) and \(-yx\) are like terms because their variables (\(xy\) and \(yx\)) are identical. Remember, in mathematics, the order of multiplication does not matter, so \(xy = yx\).
Let's look at the numerical terms first. Combine the constant numbers in the expression:

• \(-9, -14, +1\)

Perform the arithmetic:

• \(-9 - 14 = -23\)
• \(-23 + 1 = -22\)

Next are the variable terms. Combine the like terms \(-2xy\) and \(-yx\) (which is equivalent to \(-1xy\)) to get \(-3xy\).
The expression simplifies further to: \(-22 - 3xy\). Whenever you encounter like terms, don't hesitate to merge them for simplification!

Handling negative signs is crucial for correctly solving algebraic expressions. Always pay attention to their influence over terms, especially when they appear before parenthesis or within multiplication. A negative sign before a parenthesis, like in \(-(xy + 7)\), means you need to change the sign of each term inside during distribution.

• \(-xy\)
• \(-7\)

Similarly, if an expression has multiple negative numbers, combine them mindfully. For instance, merging \(-9\), \(-14\), and the positive \(+1\) requires careful arithmetic. Each negative affects the result, leading to \(-22\) after combination.
Understanding how negative signs work in different contexts within an expression helps maintain accuracy in your calculations. Whenever possible, perform small checks by manually counting negative signs to avoid mistakes.