Problem 98 - College Algebra

Question

Write each algebraic expression without parentheses. \(-(-17 y)\)

Step-by-Step Solution

Answer

The simplified form of \(-(-17 y)\) is \(17y\).

Step 1: Identify the double negatives

In the given algebraic expression \(-(-17 y)\), you can see a double negative: The negative sign in front of the parenthesis and the negative sign in front of \(17y\).

Step 2: Using the property of double negatives

The rule of double negatives states that a negative of a negative results in a positive. Therefore, we can replace \(-(-17 y)\) with \(17y\).

Key Concepts

Double Negative in Algebra

When you encounter a double negative in algebra, it can seem a bit perplexing at first, but it's actually based on a simple rule. A double negative occurs when two negative signs are next to each other, like in the expression \( -(-17y) \). This can be visually interpreted as 'negative negative seventeen y'.

To simplify, we apply a fundamental concept: two negatives make a positive. Imagine it as two wrongs making a right in the world of algebra. So, \( -(-17y) \) simplifies to \( +17y \), or just \( 17y \), since a positive sign is often implied and does not need to be explicitly shown.

Tackling Double Negatives

Here's how you can handle double negatives in your algebra problems:

• Identify the consecutive negative signs in your expression.
• Recognize that the two negatives cancel each other out and result in a positive.
• Rewrite the expression without the double negative.

Remember, simplifying an expression by removing double negatives can make your calculations much clearer and easier to follow.

Algebraic Expression Without Parentheses

Algebraic expressions are often simplified by removing parentheses, which can help clarify what you're working with. To write an algebraic expression without parentheses, you need to consider the signs and operation rules that are being applied within them.

For example, consider the expression \( -(-17y) \). Normally, a negative sign outside the parentheses indicates that you should distribute that negative sign to each term inside the parentheses. However, if there is only one term inside, it's pretty straightforward - the negative sign outside simply reverses the sign of that term.

Steps to Remove Parentheses

• Look at the term or terms inside the parentheses.
• Apply any outside operations to those terms - in this case, the negative sign flips the sign.
• Write the result without parentheses.

By doing this, you would go from \( -(-17y) \) to just \( 17y \) because you've removed the parentheses and reversed the negative sign.

Properties of Negatives in Algebra

Understanding how negatives work in algebra is crucial for correctly simplifying expressions. Negatives can reverse the value of a number or term, and when combined with other negatives, they can cancel out.

There are a few basic properties of negatives to remember:

• Negative of a Positive: The negative of a positive number is negative (\( -(+a) = -a \) ).
• Negative of a Negative: As we've seen with double negatives, the negative of a negative number is positive (\( -(-a) = +a \) or just \( a \) ).
• Multiplying Negatives: When two negative numbers are multiplied together, the result is positive (\( -a \times -b = ab \) ).
• Adding and Subtracting Negatives: Adding a negative is the same as subtracting a positive, and vice versa (\( a + (-b) = a - b \) and \( a - (-b) = a + b \) ).

These properties play a fundamental role in simplifying more complex algebraic expressions and help to understand concepts like distribution and factoring when you work with negatives.