Problem 18 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ 5(w+6) $$

Step-by-Step Solution

Answer

The simplified expression without parentheses is \(5w + 30\).

Step 1: Identify the Factor and Terms

The first step is to identify the factor and terms in the expression. In \(5(w+6)\), the factor is '5' and the terms within the parentheses are 'w' and '6'.

Step 2: Applying the Distributive Property

Now, apply the distributive property by multiplying the factor with each term in the parentheses separately. This means, multiply '5' with 'w' and also '5' with '6' separately.

Step 3: Write Down the Simplified Expression

So, the multiplication yields: \(5 * w = 5w\) and \(5 * 6 = 30\). The simplified expression by combining these results is \(5w + 30\).

Key Concepts

Multiplication

Multiplication is one of the fundamental arithmetic operations, and it's all about repeated addition. When we multiply a number by another, we are essentially adding that number to itself as many times as stated by the other number. For example, multiplying 5 by 3 is the same as adding 5 three times: 5 + 5 + 5, which equals 15.

In the context of algebra, multiplication takes on a similar role but with variables involved. Using the distributive property, we multiply a factor by each term within a set of parentheses. For example, in the expression \(5(w + 6)\), the number 5 is multiplied with both \(w\) and 6.

• First, multiply 5 by \(w\) to get \(5w\).
• Then, multiply 5 by 6 to get 30.

This multiplication by distributing the outside factor helps simplify expressions into their core components, making further calculations easier.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value. It's important in algebra to make expressions easier to work with and to solve equations. When simplifying expressions using the distributive property, each term inside a parenthesis is multiplied by the factor outside the parenthesis.

Take \(5(w + 6)\) - our goal is to remove the parentheses and simplify the expression. By applying the distributive property:

• Multiply 5 by \(w\) to obtain \(5w\).
• Multiply 5 by 6 to obtain 30.

Combine the results of these multiplications to get a simplified expression: \(5w + 30\).

Simplifying expressions is a necessary skill for solving more complex algebraic equations effectively.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operators. They represent unknown quantities and are a foundational concept in algebra. Unlike equations, algebraic expressions do not have an equals sign, which means they can't be solved but can be simplified or evaluated.

In the expression \(5(w + 6)\), we have:

• The coefficient of 5, which is a constant multiplier.
• The variable \(w\), which can represent any number.
• Another constant, which is the number 6.

Using properties like the distributive property, we can manage and manipulate expressions to form equations or to simplify them, just as we did to achieve \(5w + 30\). By understanding these expressions, students can start to model real-world scenarios with mathematical equations, opening the door to problem-solving and critical thinking in science, engineering, economics, and beyond.