Problem 51 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ -(r-3-7 p) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-r + 3 + 7p\).

Step 1: Distribute the Negative Sign

The expression is \[-(r-3-7p)\].Distribute the negative sign across the terms inside the parentheses. This means you change the sign of each term within the parentheses:\[= -r + 3 + 7p\].

Step 2: Write the Expression Without Parentheses

With the distribution completed, the expression without parentheses is:\(-r + 3 + 7p\).

Step 3: Check for Simplification

Examine the expression \(-r + 3 + 7p\) to see if any further simplification is possible. There are no like terms, so the expression is already in its simplest form.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. These expressions are fundamental in algebra and help us represent various mathematical situations and problems.
Understanding algebraic expressions is key because they allow us to model real-world situations and solve problems by applying algebraic operations like the distributive property. For example, in the expression \(-(r-3-7p)\), we have variables \(r\) and \(p\), numbers such as 3 and -7, and operations, including subtraction inside the parentheses.
The goal is to manipulate these expressions to simplify or evaluate them. By applying rules such as distributing and combining like terms, we can rewrite and understand complex expressions better.

Simplification

Simplification is the process of making an algebraic expression easier to work with by combining like terms or performing operations to reach its simplest form.
To simplify the expression \(-(r-3-7p)\), we first distribute the negative sign across all terms within the parentheses. This step transforms the expression into \(-r + 3 + 7p\). After distribution, we check for like terms or opportunities to combine terms to simplify further.
In this expression, we see that there are no like terms to combine. A like term would be a term that shares the same variable and exponent, such as \(-r\) and another term like \(2r\). Since there are none, the expression is already in its simplest form with no further simplification needed.

Negative Signs

Negative signs can sometimes be tricky in math, but they are essential for changing the direction/value of terms in an expression.
The expression \(-(r-3-7p)\) is a good example to illustrate the role of negative signs. The negative sign outside the parentheses indicates that we need to change the sign for each term inside.

• This changes \(r\) to \(-r\)
• Changes \(-3\) to \(+3\)
• And changes \(-7p\) to \(+7p\)

When the negative sign is applied correctly, it alters the terms to reflect their new values in the context of the whole expression. This process helps prevent errors in calculations and ensures accuracy in the simplification of the expression.