Problem 318 - Intermediate Algebra

Question

In the following exercises, simplify. $$ 9.6 m+7.22 n+(-2.19 m)+(-0.65 n) $$

Step-by-Step Solution

Answer

The simplified expression is \( 7.41m + 6.57n \).

Step 1: Group like terms

Identify and group like terms, which are terms that have the same variable. In the expression, the terms with 'm' are 9.6m and -2.19m, and the terms with 'n' are 7.22n and -0.65n.

Step 2: Combine like terms with variable 'm'

Combine 9.6m and -2.19m. This gives you: \[ 9.6m + (-2.19m) = 9.6m - 2.19m = 7.41m \]

Step 3: Combine like terms with variable 'n'

Combine 7.22n and -0.65n. This gives you: \[ 7.22n + (-0.65n) = 7.22n - 0.65n = 6.57n \]

Step 4: Write the simplified expression

Combine the results from Steps 2 and 3 to get the simplified expression: \[ 7.41m + 6.57n \]

Key Concepts

Like Terms

In algebra, terms are considered 'like terms' if they have identical variables raised to the same power. For example, in the expression \(9.6m + 7.22n + (-2.19m) + (-0.65n)\), the terms with 'm' (9.6m and -2.19m) are like terms, and the terms with 'n' (7.22n and -0.65n) are also like terms.

Recognizing like terms is essential in simplifying algebraic expressions because it allows you to group and combine terms efficiently. Like terms simplify into a single term that makes the expression easier to manage and solve.

Combine Like Terms

Combining like terms involves adding or subtracting their coefficients while keeping the variable part unchanged. In the example expression \(9.6m + 7.22n + (-2.19m) + (-0.65n)\), after identifying the like terms, you can combine them as follows:

• For the 'm' terms: \(9.6m + (-2.19m)\) simplifies to \(7.41m.\)
• For the 'n' terms: \(7.22n + (-0.65n)\) simplifies to \(6.57n.\)

This step-by-step combination makes the expression simpler and more compact, helping solve or further manipulate the equation.

Algebraic Simplification

Algebraic simplification is the process of making an algebraic expression as simple as possible. This involves combining like terms, reducing fractions, and eliminating unnecessary parts of the expression.

In our given example, the simplification process included identifying and combining like terms:

• We first grouped like terms: \(9.6m\) with \(-2.19m\), and \(7.22n\) with \(-0.65n\).
• We then combined these terms, leading to the simplified form \(7.41m + 6.57n.\)

The result is a cleaner and simpler expression that is easier to work with in further calculations.

Variables

Variables are symbols used to represent numbers in algebraic expressions. They allow for greater flexibility because they can stand for any number. In our example, 'm' and 'n' are the variables.

Understanding how to work with variables is crucial because:

• They let you generalize mathematical statements.
• They enable you to solve equations and inequalities.

When simplifying expressions, always keep the variable part unchanged while combining the numerical coefficients. This understanding forms the foundation of algebraic problem solving.