Problem 10 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$5 m+3 n-2 m$$

Step-by-Step Solution

Answer

Simplified expression: \(3m + 3n\).

Step 1: Identify Like Terms

Look at the expression and identify terms that are similar. The terms with \(m\) are \(5m\) and \(-2m\), as both have the variable \(m\). Although \(3n\) stands out with an \(n\) variable, it does not have like terms to combine with.

Step 2: Combine Like Terms

Combine the coefficients of the \(m\) terms: \(5m - 2m\). This means you subtract \(2m\) from \(5m\), resulting in \(3m\).

Step 3: Write the Simplified Expression

Place the term \(3m\) that you found along with the \(3n\) term from the original expression. There's nothing to simplify further for \(3n\) as it has no like terms. Thus, the simplified expression is \(3m + 3n\).

Key Concepts

Understanding Like Terms

"Like terms" is a critical concept when working with algebraic expressions. Although this might seem a bit tricky at first, it is quite straightforward once you understand the rules.
Like terms are terms within an algebraic expression that have the same variables raised to the same power. For instance, in the expression given in the exercise, you can spot the terms with the variable 'm'.
These are the terms:

• \( 5m \)
• \(-2m\)

Both are considered like terms because they have the same variable 'm'.
However, the term \(3n\) is not a like term with the others, because it contains the variable 'n'. Only terms having the same variable(s) can be combined. Recognizing this is essential as it simplifies the entire process of working through the expression.

Simplification of Algebraic Expressions

Simplification in algebra involves making an expression easier or more concise. After identifying like terms, the next step is to combine them.
For the exercise, this means we want to simplify the expression by combining the like terms we identified.
Simplification is:

• Identify like terms: Terms with the same variable(s).
• Combine these terms by adding or subtracting their coefficients.

In our exercise, you combine \(5m\) and \(-2m\). To do this:

• Subtract \(2m\) from \(5m\). Since both have 'm' as the variable, simply focus on the coefficients.

The result is:

• \(3m\)

This is then combined with the other term \(3n\), resulting in a simplified expression: \(3m + 3n\). No further simplification is possible because \(3n\) has different variables and no like terms.

The Role of Coefficients

Coefficients are the numerical parts of a term in an algebraic expression.
They tell us how many times to multiply the variable. For example, in \(5m\), the coefficient is 5. Here is how you handle coefficients:

• When combining like terms, you focus on the coefficients.
• Perform addition or subtraction by simply focusing on the coefficients. In \(5m - 2m\), you subtract 2 from 5 to get 3.

This process is the same regardless of the type of variable involved.
Even when terms are similar in terms of their variables, only those with identical variables and exponents can be combined based on coefficients. Handling coefficients correctly is crucial to effectively simplify expressions. It allows us to accurately combine like terms and arrive at the simplest form of the expression, like \(3m + 3n\) in the given exercise.