Problem 57 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4 . \(3(2 r+5)-7\)

Step-by-Step Solution

Answer

The expression simplifies to \( 6r + 8 \).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers \( a \), \( b \), and \( c \), the expression \( a(b + c) \) is equal to \( ab + ac \). Apply this property to the expression \( 3(2r + 5) \). This gives \( 3 \times 2r + 3 \times 5 = 6r + 15 \).

Step 2: Write Expression Without Parentheses

Substitute the result from Step 1 back into the original expression. The expression now is \( 6r + 15 - 7 \) without the parentheses.

Step 3: Simplify the Expression

Combine like terms in the expression from Step 2. Subtract \(7\) from \(15\) to simplify the constant terms: \( 15 - 7 = 8 \). Thus, the simplified expression is \( 6r + 8 \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing algebraic expressions to their most compact form. This means performing all possible arithmetic operations. It also includes combining any like terms that appear in the expression after utilizing properties such as the distributive property. In the exercise provided, after applying the distributive property and opening up the parentheses, we dealt with the expression \(6r + 15 - 7\). Here, simplifying it means we need to reduce the constants present. So, subtract 7 from 15, resulting in the simplified expression \(6r + 8\). This is a basic example of expression simplification which helps prepare the way for solving more complex equations. Remember to always perform all possible operations to simplify your expressions as much as possible.

Algebraic Expressions

Understanding algebraic expressions is central to succeeding in algebra. An algebraic expression is a mathematical phrase that can include numbers, variables (like \(r\) in this example), and operation symbols. Unlike equations, they do not have an equality sign. In the given task, the expression started as \(3(2r+5) - 7\). The purpose is to manipulate this expression into a simpler form, eliminating the parentheses, by using algebraic rules and operations. Algebraic expressions can represent real-world quantities, where variables stand in for unknown or varying values. Whether you’re dealing with simple expressions or more complex ones, understanding the structure and components of algebraic expressions is fundamental to applying necessary operations and transformations effectively.

Like Terms

Like terms are an important concept when simplifying expressions. Like terms refer to terms in an algebraic expression that have identical variable parts, though their coefficients can differ. In our exercise, we identify like terms in \(6r + 15 - 7\). The variable term \(6r\) is standalone, but when simplifying the expression, \(15\) and \(-7\) are like terms. These are constant terms that we can combine by performing arithmetic operations. Using like terms helps simplify expressions by combining the separable parts into manageable pieces, making complex expression manipulations more approachable. Properly identifying like terms is crucial as it allows for the reduction of expressions to their simplest form. Always keep an eye on those terms that share variables or that are part of the constant numerical operations within the expression.