Problem 370 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 8-(r-7) $$

Step-by-Step Solution

Answer

15 - r

Step 1: Identify the terms within the parentheses

First, identify the terms inside the parentheses that need to be distributed. Here, the expression inside is \(r - 7\).

Step 2: Apply the Distributive Property

Using the Distributive Property, distribute the negative sign through the parentheses: \(8 - (r - 7) = 8 - r + 7\).

Step 3: Simplify the expression

Combine the like terms of the expression to simplify: \(8 + 7 - r = 15 - r\).

Key Concepts

Simplifying algebraic expressions

Simplifying algebraic expressions means making the expression as simple as possible. This often involves removing parentheses and combining like terms. For example, in the given exercise, we start with the expression \[8 - (r - 7)\]To simplify this, we need to apply the Distributive Property and then combine like terms to get the final simplified expression. Simplifying helps in easier calculation and understanding of the expression. It’s a crucial skill in algebra, as it forms the basis of solving equations and inequalities.

Combining like terms

Combining like terms means adding or subtracting terms that have the same variable raised to the same power. Like terms can be constants or variables. In our expression \[8 - r + 7\]The like terms are the constants \(8\) and \(7\). When we combine these, we get \[15 - r\]Always be on the lookout for like terms in any algebraic expression. This step makes the expression simpler and easier to work with. Remember, you can only combine terms that are exactly alike. For instance, you can't combine \(r\) and \(r^2\) because they are not like terms.

Negative sign distribution

Distributing the negative sign correctly is essential to avoid mistakes in simplifying expressions. In our exercise \[8 - (r - 7)\]we need to distribute the negative sign to each term inside the parentheses. Essentially, you're multiplying each term inside the parentheses by \(-1\). Thus \[-(r - 7)\]becomes \[-r + 7\]So, the original expression \[8 - (r - 7)\]transforms into \[8 - r + 7\]Afterwards, combining like terms gives the final simplified form as \[15 - r\]Always remember to apply the negative sign to each term inside the parentheses. This prevents any errors and ensures an accurate simplification.