- Simplify the first expression: \(2m + 5 + m - 4 = 3m + 1\).
- Simplify the second expression: \(-5m - 4 + 8m + 5 = 3m + 1\).
- Since both expressions simplify to the same polynomial, they are equivalent for all values of \(m\).
- The best description is that they are equivalent because they yield the same value at \(m = 4\). The final answer is \(\boxed{\text{They are equivalent because when } m = 4, \text{ the two expressions have the same value.}}\)
Explanation
- Simplifying and Comparing the Expressions
To determine if the two expressions are equivalent, we simplify them individually by combining like terms.
For the first expression:
\[ 2m + 5 + m - 4 \]
We group the terms containing \(m\) and the constant terms:
\[ (2m + m) + (5 - 4) \]
\[ = 3m + 1 \]
For the second expression:
\[ -5m - 4 + 8m + 5 \]
We group the terms containing \(m\) and the constant terms:
\[ (-5m + 8m) + (-4 + 5) \]
\[ = 3m + 1 \]
Since both expressions simplify to the same polynomial, \(3m + 1\), they are equivalent for all values of \(m\).
- Evaluating the Options
Now we evaluate the provided options:
- "They are equivalent because when \(m = 4\), the two expressions have the same value." - Since the expressions are equivalent for all \(m\), they must have the same value at \(m = 4\). This is a true statement, although it only demonstrates equivalence at a single point.
- "The two expressions are not equivalent because when they are simplified, they do not have the same coefficient for the \(m\) term." - This is incorrect because both expressions simplify to have a coefficient of 3 for the \(m\) term.
- "They are equivalent because the sum of the constants is the same in both expressions." - This is an incomplete justification because equivalence requires both the variable and constant terms to match.
- "The two expressions are not equivalent because when \(m = 4\), the two expressions do not have the same value." - This is incorrect because the expressions are equivalent and thus have the same value at \(m=4\).
The first option correctly identifies that they are equivalent and cites a correct (if partial) reason. Given the choices, this is the best description.
- Conclusion
By simplifying both expressions, we have shown that they are equivalent because they both reduce to \(3m + 1\). Comparing this to the given choices, the statement "They are equivalent because when \(m = 4\), the two expressions have the same value" is the most accurate description. Although equivalence is a property for all \(m\), this choice correctly asserts equivalence and provides a valid test case as evidence. The final answer is the first option.
Examples
In business or engineering, simplifying expressions is akin to optimizing processes. If two different workflows (expressions) result in the same output (value) regardless of the input (variable \(m\)), they are equivalent. Recognizing this equivalence allows individuals to choose the simplest or most efficient method to achieve a goal. For example, if two payment plans for a service simplify to the same total cost over time, a customer can confidently choose either, knowing they are mathematically identical.