Problem 26 - Algebra A Combined Function

Question

Simplify each expression. Use the distributive property to remove any parentheses. $$ 7(r+3) $$

Step-by-Step Solution

Answer

The simplified expression is \(7r + 21\).

Step 1: Use the Distributive Property

Apply the distributive property to the expression \(7(r+3)\). According to the distributive property, \(a(b+c) = ab + ac\). In this case, \(a = 7\), \(b = r\), and \(c = 3\).

Step 2: Distribute 7 to Each Term

Multiply each term inside the parentheses by 7. This means you perform two separate multiplications: \(7 \times r\) and \(7 \times 3\).

Step 3: Simplify Each Product

Compute the products: \(7 \times r = 7r\) and \(7 \times 3 = 21\).

Step 4: Combine the Results

Add the results from the distributed terms together. This gives you \(7r + 21\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing complex expressions into their simplest form. The goal is to make them easier to work with or solve. To simplify an expression, particularly when it includes parentheses, you typically:

• Apply the distributive property to eliminate parentheses.
• Combine like terms, if possible.

In the given exercise, the expression is simplified by using the distributive property, eliminating the parentheses, and then performing the basic arithmetic that follows. Simplifying helps provide clarity and prepares the equation for additional operations or evaluations. This process is foundational in algebra and helps build intuition for manipulating algebraic expressions efficiently.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables (like letters), and operation symbols. Unlike an equation, it does not have an equal sign. Algebraic expressions form the basis for building equations and inequalities.

• Variables like \(r\) can represent unknown values or quantities that can change.
• Constants like \(3\), when not part of a variable term, usually hold fixed numerical values.

In the expression \(7(r+3)\), \(r\) is the variable, while \(7\) and \(3\) are constants. Algebraic expressions allow you to create general formulas that can be applied to multiple situations, making them versatile tools in problem-solving contexts.

Multiplication in Algebra

Multiplication in algebra follows similar rules to arithmetic multiplication, but it often involves variables and unknowns. The distributive property is a core principle when dealing with multiplication in algebra. This property allows you to multiply a single term by each term inside a set of parentheses.

• This method helps in breaking down complex expressions into simpler parts.
• For example, multiplying \(7(r+3)\) results in each term within the parentheses being multiplied separately by \(7\): \(7 \times r\) and \(7 \times 3\).

By mastering these concepts, you can handle expressions with ease and transform them into more manageable forms. Remember that performing these steps in order makes the process efficient and helps maintain accuracy in your calculations.