Problem 51 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 0.5(m+2)+0.4 m $$

Step-by-Step Solution

Answer

The simplified expression is \(0.9m + 1\).

Step 1: Distribute the Coefficient

First, apply the distributive property which states that multiplying a number by a sum is the same as doing each multiplication separately. This means you multiply the 0.5 across the terms inside the parentheses: \(0.5(m + 2) = 0.5m + 0.5 \times 2\). Calculate \(0.5 \times 2\) to get 1. So, the expression becomes \(0.5m + 1 + 0.4m\).

Step 2: Combine Like Terms

Now, identify and combine the like terms in the expression. The like terms here are the terms that contain \(m\), which are \(0.5m\) and \(0.4m\). Add these terms together: \(0.5m + 0.4m = 0.9m\). Therefore, the expression now simplifies to \(0.9m + 1\).

Key Concepts

Combining Like Terms

Combining like terms is a core concept in algebra that helps simplify expressions. It involves grouping and adding or subtracting terms that have the same variables and exponents. This technique allows for a more streamlined equation.

• Look for terms with the same variable, such as \(m\) in our example.
• Add or subtract their coefficients. For example, \(0.5m + 0.4m\) equals \(0.9m\).

To combine like terms, first ensure that the terms are alike by comparing their variable parts. Only terms that have identical variables and powers can be combined. This process reduces the number of terms and simplifies the expression substantially, making calculations easier.

Simplifying Expressions

Simplifying expressions is about rewriting them in the simplest form, which often involves combining like terms and performing any basic arithmetic operations. This makes the expressions easier to understand and work with.

• Start with removing any unnecessary parentheses.
• Perform multiplication and division operations.
• Combine like terms.

In our example, after distributing the coefficients and removing parentheses, we simplify the expression to \(0.9m + 1\). This final form is as straightforward as it gets, making it easy to read and further use in problems.

Removing Parentheses

Removing parentheses is often the first step in simplifying expressions. Parentheses indicate multiplication by a factor out front, which needs to be distributed across the terms inside.Removing parentheses allows you to apply other simplification strategies. Here's how it works:

• Apply the distributive property: multiply the factor outside by each term inside the parentheses.
• For example, distribute \(0.5\) in \(0.5(m + 2)\) to get \(0.5m + 1\).
• Once the multiplication is done, the parentheses are gone, and you can move on to combining like terms.

This process ensures that each term is treated properly and paves the way for further simplification. It’s a vital step in transforming complex expressions into simple, manageable ones.