Problem 69 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ 7(t+3) $$

Step-by-Step Solution

Answer

7t + 21

Step 1: Identify the components

Recognize that the expression is in the form of a product of a single term and a binomial: This can be written as: \[7(t+3)\]

Step 2: Apply the distributive property

The distributive property states that \(a(b + c) = ab + ac\). In this case, the term \(7\) needs to be distributed across the binomial \((t + 3)\): \[7(t) + 7(3)\]

Step 3: Simplify the expression

Now, perform the multiplication for each term: \[7(t) = 7t\] and \[7(3) = 21\] Combine the terms to get: \[7t + 21\]

Key Concepts

Multiplication

Multiplication is a fundamental mathematical operation that combines two numbers to yield a third number. In the context of algebra, multiplication often involves combining constants with variables. Here, we use multiplication to distribute a single number across a binomial. When you see an expression like \(7(t+3)\), you're meant to multiply 7 by each part of the binomial. This means: \[7 \times t \text{ and } 7 \times 3.\] Understanding this basic concept will help you simplify more complex expressions.

Binomial

A binomial is an algebraic expression containing exactly two terms. It takes the form \(a + b\). In our exercise, the binomial is \(t + 3\). Binomials often appear in expressions that use the distributive property of multiplication. Recognizing a binomial is straightforward: look for two terms separated by a plus \(+\) or minus \(-\) sign. Knowing this helps us to apply multiplication correctly.

Simplification

Simplification is the process of reducing an expression to its simplest form. After applying the distributive property, we are left with smaller, simpler expressions that are easier to understand. Here, we've taken our initial expression \(7(t+3)\) and distributed the 7 across both terms to get: \[7 \times t + 7 \times 3 = 7t + 21.\] This is the simplified form where multiplication has been resolved, yielding an easily understandable expression. Simplification is crucial in solving algebraic problems as it makes complex expressions more manageable and sets you up for solving further equations down the line.