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Chapter 2: Problem 59

Simplify each expression. See Examples 3 through 7 . $$ 0.5(m+2)+0.4 m $$

Short Answer

Expert verified
The expression simplifies to \( 0.9m + 1 \).

Step by step solution

01

Distribute the Coefficient

Starting with the expression \( 0.5(m+2)+0.4m \), distribute \( 0.5 \) to both \( m \) and \( 2 \) inside the parentheses. This gives us \( 0.5m + 0.5 \times 2 \). Calculate \( 0.5 \times 2 \) to get 1. So the expression becomes \( 0.5m + 1 + 0.4m \).
02

Combine Like Terms

Now combine the like terms in the expression \( 0.5m + 1 + 0.4m \). The like terms are \( 0.5m \) and \( 0.4m \). Add them together: \( 0.5m + 0.4m = 0.9m \).
03

Simplification

After combining like terms, the expression reduces to \( 0.9m + 1 \). This is the simplest form of the original expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental tool in algebra that allows you to simplify expressions involving parentheses. When you see a coefficient outside parentheses followed by an expression inside, you need to distribute the coefficient to each term within the parentheses.

In the exercise, we start with the expression \( 0.5(m+2)+0.4m \). The coefficient \( 0.5 \) is distributed to both \( m \) and \( 2 \). This means you multiply \( 0.5 \) by both \( m \) and \( 2 \).
  • First, calculate \( 0.5 \times m \) to get \( 0.5m \).
  • Next, calculate \( 0.5 \times 2 \) to get \( 1 \).
Putting it together, the expression becomes \( 0.5m + 1 + 0.4m \). By distributing the number correctly, you break down the expression into easier steps for further simplification.
Combining Like Terms
After using the distributive property, you'll be left with an expression that may include like terms. Like terms are terms that have the same variable raised to the same power. You can simply add or subtract the coefficients of these terms to combine them.

In the expression \( 0.5m + 1 + 0.4m \), notice the like terms: \( 0.5m \) and \( 0.4m \). They both have the variable \( m \), so you can combine them.
  • Add the coefficients: \( 0.5 + 0.4 \).
  • This gives you \( 0.9m \).
Now, the expression simplifies to \( 0.9m + 1 \). By combining like terms, you make the expression easier to work with, eliminating extra terms and creating a cleaner equation.
Algebraic Simplification
Algebraic simplification is the process of transforming an expression into its simplest form by using various algebraic techniques. Often, this involves applying the distributive property and combining like terms, as we did here. The goal is to make the expression as straightforward as possible.

Starting with a complex expression, you use steps that simplify it to something concise and clear, like \( 0.9m + 1 \). With simplification:
  • You perform calculations to eliminate complex parts.
  • Reduce the number of terms by combining those that are alike.
Algebraic simplification is essential for solving equations, as it allows you to see clearly the relationships between variables. It is also crucial in making expressions more manageable when working with them in further mathematical operations.

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