Problem 44 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. \(-5(2 r+11)\)

Step-by-Step Solution

Answer

The simplified expression is \(-10r - 55\).

Step 1: Apply the Distributive Property

The expression given is \(-5(2r + 11)\). To remove the parentheses, apply the distributive property by multiplying \(-5\) with each term inside the parentheses: \(-5 \times 2r\) and \(-5 \times 11\).

Step 2: Perform the Distributive Multiplication

Calculate each multiplication step separately: \(-5 \times 2r = -10r\) and \(-5 \times 11 = -55\).

Step 3: Combine the Results

Combine the results from the multiplication: \(-10r - 55\). This is the simplified version of the original expression.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators like addition, subtraction, multiplication, and division. In these expressions, numbers are known as coefficients, while variables represent unknown values. For example, in the expression \(2r + 11\), \(2\) is the coefficient, and \(r\) is the variable that can take different numerical values. The number \(11\) is known as a constant term since its value does not change.

Understanding algebraic expressions is essential because they form the building blocks for solving equations in algebra. They allow us to model real-world situations mathematically and find solutions to various problems.

• Coefficients are numbers that multiply the variables.
• Variables are symbols that represent unknown quantities.
• Constants are fixed numbers.

Grasping these basics is key to progressing in algebra.

Simplification of Algebraic Expressions

Simplification refers to the process of reducing an algebraic expression to its simplest form. The goal is to make the expression easy to understand and use. In the context of the distributive property, simplification often involves eliminating parentheses and combining like terms, as seen in the exercise. This process allows us to work more efficiently with the expressions in calculations and problem-solving.

In the given expression \(-5(2r+11)\), after applying the distributive property, we obtained \(-10r - 55\). Here, no further simplification is needed since there are no like terms to combine. This expression is now in its simplest form.

• The simplification process can involve multiple steps, such as applying operations and combining terms.
• "Like terms" are terms that have the same variables raised to the same power.
• A simplified expression is more concise and easier to interpret in mathematical problems.

The Role of Multiplication in Distributive Property

Multiplication plays a crucial role in the simplification of algebraic expressions, especially when using the distributive property. The distributive property states that a term outside the parentheses multiplies each term inside the parentheses. For example, in \(-5(2r + 11)\), \(-5\) is multiplied by both \(2r\) and \(11\). This step eliminates the parentheses and simplifies further operations.

The distributive property is a powerful tool because it allows us to simplify expressions and equations that would otherwise be complex. Proper application of multiplication in this way lays the foundation for accurate calculations in algebra and even more advanced math topics.

• The distributive property is expressed as \(a(b+c) = a \times b + a \times c\).
• It simplifies expressions by breaking them down into smaller, more manageable parts.
• Understanding how to distribute multiplication across terms in parentheses is key to mastering algebraic manipulation.

Mastering these techniques will help significantly in tackling a wide range of mathematical problems more effectively.