Problem 66 - Intermediate Algebra

Question

Simplify by combining like terms. See Example 5 . $$-5.7 m+5.3 m$$

Step-by-Step Solution

Answer

-0.4m

Step 1: Identify Like Terms

The expression given is \(-5.7m + 5.3m\).Both terms are like terms since they have the same variable part \(m\). Thus, they can be combined by addressing their coefficients.

Step 2: Combine the Coefficients

To combine the like terms, add the coefficients of \(m\). The coefficients here are \(-5.7\) and \(5.3\). Compute: \(-5.7 + 5.3\) which is \(-0.4\).

Step 3: Formulate the Simplified Expression

Multiply the resultant coefficient by the variable \(m\). Thus, the expression becomes\(-0.4m\).

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that lets you break down complex problems into simpler, more manageable forms. When we simplify an algebraic expression, we make it easier to understand or solve by reducing it to its most basic form without changing its value. For example, consider the expression \(-5.7m + 5.3m\). This can potentially look a little intimidating at first glance, especially with decimal numbers involved. But simplification can make it much simpler.Here's how it works:

• Identify Like Terms: Look for terms in the expression that have the same variable with the same exponent. For instance, \(-5.7m\) and \(5.3m\) are both terms that include the same variable \(m\).
• Combine: Add or subtract these like terms by addressing their coefficients. It’s easy once you’ve recognized the like terms—just based on basic addition or subtraction of numbers.

This approach simplifies our original expression to \(-0.4m\), making it easier to handle in further calculations or when solving equations.

Coefficients

In algebra, coefficients are numbers that multiply the variables in an expression. Knowing how to work with coefficients is important when you're simplifying expressions or solving equations. Here's a bit more on coefficients:

• Understand What They Represent: In the expression \(-5.7m\), \(-5.7\) is the coefficient. It tells you how many times the variable \(m\) is being multiplied.
• The Role in Simplification: When combining like terms, you're really manipulating these coefficients. So in \(-5.7m + 5.3m\), the operation boils down to combining the coefficients: \(-5.7 + 5.3 = -0.4\).

By focusing on coefficients, you streamline the process of simplifying expressions, keeping the variable part intact, and making calculations more straightforward.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition and subtraction). Understanding their structure helps you manipulate them more effectively, such as when simplifying or solving them. Let's break down what they typically involve:

• Variables: These are symbols (often letters) that represent unknown numbers. For instance, in the expression \(-5.7m + 5.3m\), \(m\) is the variable.
• Terms: Each part of an algebraic expression separated by a plus or minus sign is a term. Here, \(-5.7m\) and \(5.3m\) are individual terms.
• Operators: These include signs like + or - that define how the terms are used together.

Recognizing these components is crucial for simplifying expressions or tackling algebraic equations confidently. Each part plays a role in defining the expression's overall structure and function.