Problem 53 - Fundamentals of Mathematics

Question

For problems \(47-56\), simplify each expression by combining like terms. $$ 0 m+3 k-5 s+2 m-s $$

Step-by-Step Solution

Answer

The simplified expression is \( 2m + 3k - 6s \).

Step 1: Identify Like Terms

To simplify the given expression, we first need to identify the like terms. Like terms are terms that contain the same variables raised to the same power. In the expression: \[ 0m + 3k - 5s + 2m - s \]we recognize the like terms: - Terms involving \( m \) are \( 0m \) and \( 2m \)- Terms involving \( s \) are \( -5s \) and \( -s \)- \( 3k \) has no like terms.

Step 2: Combine Like Terms

Combine the like terms identified in Step 1 as follows:- For \( m \) terms: \( 0m + 2m = 2m \)- For \( s \) terms: \( -5s - s = -6s \)- The term \( 3k \) remains unchanged because it has no like terms.Thus, the simplified expression is:\[ 2m + 3k - 6s \]

Key Concepts

Understanding Like Terms

Like terms are an essential part of algebraic expressions. They make simplifying expressions more straightforward. At their core, like terms are terms within an expression that have identical variable parts.
When we look at an expression like \( 0m + 3k - 5s + 2m - s \), we identify like terms by checking these factors:

• They must contain the same variable, such as both being terms in \( m \) or \( s \).
• The exponent of these variables should also match. For example, \( m^2 \) is not like \( m \).

In our example:

• The \( m \) terms include \( 0m \) and \( 2m \), while the \( s \) terms are \( -5s \) and \( -s \).
  The term \( 3k \) stands alone without any like terms to pair with in this expression.

Steps to Simplifying Expressions

Simplification makes algebraic expressions easier to read and use. By simplifying, we reduce expressions to their simplest form.
Follow these steps to simplify effectively:

• First, identify like terms by matching variable parts as discussed.
• Next, group the like terms together. This doesn't change the expression's meaning but organizes it for ease of calculation.
• Finally, perform mathematical operations, like addition or subtraction, on the coefficients of the like terms.

For our expression:

• We regroup \( m \) terms as \( 0m + 2m \) to get \( 2m \).
• Regroup \( s \) terms as \( -5s - s \), which simplifies to \( -6s \).
• In the case of \( 3k \), since no grouping occurs, it remains the same.

As a result, the expression becomes \( 2m + 3k - 6s \). This is now simplified.

Combining Like Terms Effectively

Combining like terms is a powerful technique to condense algebraic expressions. This technique reduces clutter by merging similar terms.
Here's how to combine them smoothly:

• First, ensure that terms being combined truly have matching variables and exponents. Mismatched terms cannot be combined.
• Add or subtract the coefficients of these similar terms. For instance, with terms \( 2m \) and \(-m\), you would adjust coefficients to get \( m \).

Using our primary example:

• The terms \( 0m + 2m \) combine because their coefficients are added, resulting in \( 2m \).
• The terms \( -5s \) and \( -s \) are combined to yield \( -6s \).
• The term \( 3k \), because isolated, remains unaffected during this process.

Combining like terms offers clarity and streamlines algebraic calculations, preparing expressions for more complex operations. This adjustment produces a cleaner expression: \( 2m + 3k - 6s \).