Problem 72 - College Algebra

Question

simplify each algebraic expression. $$ -(5 x-13 y-1) $$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression \( -(5x - 13y - 1) \) is \( -5x + 13y + 1 \).

Step 1: Distribute The Negative Sign

The task is to distribute the negative sign to each term within the brackets. This means that each term's sign will be flipped. So, the negative sign is distributed to \( 5x \), \( -13y \), and \( -1 \). The result is \( -5x + 13y + 1 \).

Step 2: Confirm The Simplification

Now that the negative sign has been distributed, the expression is simplified as there are no more operations that can be performed on it. Hence, \( -(5x - 13y - 1) \) simplifies to \( -5x + 13y + 1 \).

Key Concepts

Simplification

Simplification in algebra involves making expressions more straightforward and easier to work with. It's like tidying up a room by organizing everything into its right place. When dealing with an algebraic expression, simplification usually consists of combining like terms and reducing the expression to a form where no further operations can be performed.
This makes equations simpler and more convenient to handle in calculations or when solving equations.

• Identify similar terms: Look for terms that can be combined. They usually share the same variable and exponent.
• Perform operations: Combine or reduce the terms wherever possible.

In our example, the main simplification step was distributing the negative sign correctly. Once you've done this, and all terms are adjusted, the expression is as simple as it can get without additional context.

Distribution Property

The distribution property is a fundamental concept in algebra. It allows you to multiply a single term across terms within parentheses, spreading the multiplication process. This is especially useful when variables and constants are grouped together.
The distribution property states:

• If you have an expression of the form \( a(b + c) \), you can distribute \( a \) by multiplying it with each term inside the parenthesis. This gives you \( ab + ac \).

In our example:

• The expression \(-(5x - 13y - 1)\) requires the "-" to be distributed. Thus, the "-" is multiplied with each of the terms inside the parentheses.
• The operation is handled sequentially, ensuring each term is affected, resulting in \( -5x + 13y + 1 \).

This operation shows how a simple minus sign outside a parenthesis can fundamentally change each term within it.

Negative Signs

Handling negative signs is crucial in algebra. They can change the value and outcome of an expression significantly. A negative sign in front of a parenthesis indicates you need to reverse the sign of every term inside the parenthesis.
Think of it like flipping a coin over to display the opposite side:

• Positive becomes negative: If a term is positive, adding a negative sign makes it negative.
• Negative becomes positive: If a term is negative, a negative sign in front will make it positive.

In our example, each term inside \( -(5x - 13y - 1) \) underwent sign changes:

• \( 5x \) became \( -5x \)
• \( -13y \) became \( +13y \)
• \( -1 \) became \( +1 \)

Understanding negative signs drastically alters algebraic thinking, making expression transformations easier to manage.

Expression Simplification

Expression simplification combines all the concepts we've discussed: simplification, distribution, and handling negative signs. This method requires practice but it makes solving problems much more efficient.

• Evaluate the expression: Look at what operations need performing. This includes distributing terms or flipping negative signs.
• Perform the steps: Distribute negatives first and then simplify by combining like terms.
• Review the expression: Ensure no further simplification is possible. Your end goal is to reduce complexity.

In our specific case, we performed simplification by distributing the negative and confirming the process afterward. The transformed expression \(-5x + 13y + 1\) is the simplified state of \(-(5x-13y-1)\). Simplicity and accuracy in expression simplification allow for clearer result interpretation and easier problem solving.