Problem 29 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -(y-9) $$

Step-by-Step Solution

Answer

-y+9

Step 1: Distribute negative sign

Distribute the '-' sign across the contents of the parentheses. This means multiplying each term in the parentheses by '-1'. Hence, we get \(-y - (-9)\), which can be more clearly read as \(-1*y + -1*-9\).

Step 2: Simplify the expression

Now, simplify the expression. We keep \(-1*y\) as \(-y\), and since a negative times a negative equals a positive, we have \(-1*-9\) becoming \(+9\). Combined, we get \(-y+9\).

Key Concepts

Negative Numbers

Negative numbers may seem tricky, but they're just numbers below zero. They have unique rules when it comes to operations like addition, subtraction, and multiplication. A key point is that multiplying two negative numbers gives a positive result. For example,

• The multiplication of \(-1\times-9\) results in \(+9\).
• It’s like saying, 'a minus of a minus is a plus.'

When dealing with expressions like \(-(y-9)\), we distribute the negative sign across the terms inside the parentheses. Each term inside is multiplied by \(-1\), which changes the sign of each term.

Simplifying Expressions

Simplifying expressions involves performing operations to make an expression as straightforward as possible. This process often includes eliminating parentheses and combining like terms. Here’s how it works:

• First, distribute any constants or variables outside the parentheses.
• In \(-(y-9)\), the negative sign is distributed to both \(y\) and \(-9\).
• Resulting in \(-y\) from \(-1\times y\) and \(+9\) from \(-1\times-9\).

Once distributed, you combine any like terms. Since there are no like terms here, \(-y+9\) is the simplified expression.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations working together. They are like mathematical phrases that can represent everything from simple to complex relationships. In \(-(y-9)\), we’re working with an algebraic expression that includes a variable \(y\).

• Variables represent unknown values and can take different numbers as you solve equations.
• Expressions are simplified using properties like the distributive law, which helps break down complex expressions into easier-to-handle parts.
• Remember, to treat each part of the expression following mathematical rules such as making a negative positive when multiplied by another negative.

Understanding how to manipulate expressions is crucial in algebra and helps in solving equations and real-life problems.