Problem 52 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 0.2(k+8)-0.1 k $$

Step-by-Step Solution

Answer

The simplified expression is \(0.1k + 1.6\).

Step 1: Distribute the Coefficient Inside the Parentheses

First, we need to remove the parentheses by distributing the 0.2 to both terms inside the parentheses. The expression inside the parentheses is \( k + 8 \). So, \( 0.2(k + 8) = 0.2 \cdot k + 0.2 \cdot 8 \).This simplifies to \( 0.2k + 1.6 \).

Step 2: Simplify the Expression by Combining Like Terms

Now that we've removed the parentheses, substitute the expression back into the original equation: \( 0.2k + 1.6 - 0.1k \)To simplify, combine the \( k \) terms: \( 0.2k - 0.1k = 0.1k \).Thus, the expression simplifies to \( 0.1k + 1.6 \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that deals with how to handle expressions with parentheses. It helps simplify expressions by distributing a multiplication over addition or subtraction inside parentheses. Think of it as handing out the multiplication to each term inside the parentheses.
To apply the distributive property, follow these steps:

• Identify the term outside the parentheses that needs to be distributed.
• Multiply that term by each term inside the parentheses.
• Simplify the result by completing the multiplication.

In our example, the distributive property is used when 0.2 is multiplied by both \( k \) and 8 in the expression \( 0.2(k + 8) \). This results in \( 0.2 \cdot k + 0.2 \cdot 8 \), simplifying to \( 0.2k + 1.6 \). By distributing, we've eliminated the parentheses and prepared the expression for further simplification.

Like Terms

In algebra, identifying and working with like terms is crucial for simplifying expressions. Like terms are terms within an expression that have the same variables raised to the same power. Only the coefficients of these terms can differ and can be combined through addition or subtraction.
Identifying like terms helps simplify complex expressions:

• Look for terms that contain the same variable(s).
• Combine the coefficients of these terms while keeping the variable part unchanged.

In the example \( 0.2k + 1.6 - 0.1k \), the like terms are \( 0.2k \) and \( -0.1k \) because both contain the variable \( k \). These can be combined to give \( 0.1k \). By combining like terms, we shorten and clarify the expression, making it easier to interpret and solve.

Simplifying Expressions

Simplifying expressions is a common task in algebra, allowing you to rewrite expressions in their most efficient form. It involves several steps, including applying the distributive property and combining like terms.
The process is all about reducing the expression to make it more manageable:

• First, resolve any operations within parentheses using the distributive property.
• Next, identify and combine like terms.
• Simplify the coefficients and constants to reduce the expression to its simplest form.

In this example, after applying the distributive property and identifying like terms, the expression \( 0.2(k + 8) - 0.1k \) simplifies to \( 0.1k + 1.6 \). This is a cleaner form that accurately represents the original expression but is easier to use in further calculations.