Select All Of The Expressions Equivalent To (0.5p + 7) + (0.3p - 6). A. 1 + 0.8p B. -6.5p + 7.3 C. 7.5p - 5.7 D. 0.8p
Understanding the Problem: Finding Equivalent Expressions
Mathematics often involves simplifying algebraic expressions to identify equivalent forms. The problem at hand asks us to determine which of the provided options are equivalent to the expression:
\[
(0.5p + 7) + (0.3p - 6)
\]
This requires a thorough understanding of combining like terms, simplifying expressions, and verifying equivalence through algebraic manipulation.
Step 1: Simplify the Given Expression
Before comparing the options, let's first simplify the expression:
\[
(0.5p + 7) + (0.3p - 6)
\]
This involves adding the two binomials, which is essentially combining like terms:
- Combine the coefficients of \(p\):
\[
0.5p + 0.3p = 0.8p
\]
- Combine the constants:
\[
7 - 6 = 1
\]
Thus, the simplified form of the original expression is:
\[
0.8p + 1
\]
Step 2: Analyze Each Option to Check for Equivalence
Now that we know the simplified expression is \(0.8p + 1\), we can compare it to each of the provided options:
- Option A: 1 + 0.8p
- Option B: -6.5p + 7.3
- Option C: 7.5p - 5.7
- Option D: 0.8p
Let's evaluate each.
Option A: 1 + 0.8p
- This is algebraically identical to \(0.8p + 1\), just written in a different order.
- Conclusion: It is equivalent to the simplified form.
Option B: -6.5p + 7.3
- This has different coefficients and constants compared to \(0.8p + 1\).
- Conclusion: Not equivalent.
Option C: 7.5p - 5.7
- Significantly different coefficients and constants.
- Conclusion: Not equivalent.
Option D: 0.8p
- Lacks the constant term \(+1\), so it's not the same as the simplified expression.
- Conclusion: Not equivalent.
Step 3: Final Selection of Equivalent Expressions
Based on the comparison:
- Option A matches the simplified form exactly.
- Options B, C, D do not match the simplified expression.
Therefore, the only equivalent expression is Option A: 1 + 0.8p.
Additional Insights: Why Recognizing Equivalent Expressions Matters
Understanding how to recognize equivalent expressions is an essential skill in algebra. It enables students and professionals to:
- Verify solutions in algebraic equations.
- Simplify complex expressions for easier manipulation.
- Identify equivalent forms for problem-solving efficiency.
- Understand deeper relationships between algebraic expressions.
Common Mistakes When Comparing Algebraic Expressions
While comparing expressions, students often make errors such as:
- Ignoring the order of terms, especially with addition, since it doesn't affect the sum.
- Confusing coefficients and constants, leading to incorrect conclusions.
- Overlooking the importance of signs (+ or -) in constants and coefficients.
- Not simplifying expressions fully before comparison.
To avoid these mistakes:
- Always simplify both expressions completely.
- Write expressions in a standard form: constants and like terms grouped.
- Double-check the coefficients and constants after simplification.
Practical Applications of Simplifying Algebraic Expressions
The ability to simplify and compare algebraic expressions is crucial across many fields, including:
1. Engineering
- Simplifying complex formulas to optimize calculations.
- Verifying the equivalence of different circuit equations.
2. Computer Science
- Optimizing algorithms by simplifying expressions.
- Validating code that involves algebraic computations.
3. Economics and Finance
- Simplifying financial formulas to analyze economic models.
- Comparing different profit or cost functions for equivalence.
4. Education
- Building foundational skills in algebra.
- Developing problem-solving strategies.
Summary: How to Identify Equivalent Algebraic Expressions
To determine whether two expressions are equivalent:
- Rewrite both expressions in standard form, combining like terms.
- Compare coefficients of similar terms.
- Compare constant terms.
- If all corresponding terms match, the expressions are equivalent.
- Use substitution if necessary: pick a specific value for \(p\) and see if both expressions yield the same result.
Example: Using substitution for verification.
Suppose \(p = 2\):
- Original simplified expression: \(0.8(2) + 1 = 1.6 + 1 = 2.6\)
- Option A: \(1 + 0.8(2) = 1 + 1.6 = 2.6\)
Both yield 2.6, confirming equivalence.
Test with another value, e.g., \(p = -1\):
- Original: \(0.8(-1) + 1 = -0.8 + 1 = 0.2\)
- Option A: \(1 + 0.8(-1) = 1 - 0.8 = 0.2\)
Again, they match, confirming that Option A is equivalent.
Conclusion: The Correct Answer
In conclusion, after simplifying the given expression and analyzing each option, the only expression equivalent to \((0.5p + 7) + (0.3p - 6)\) is Option A: 1 + 0.8p.
This exercise highlights the importance of algebraic simplification, careful comparison, and verification through substitution. Mastering these skills enhances problem-solving efficiency and deepens understanding of algebraic relationships, which are foundational in higher mathematics, science, engineering, and many other fields.
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Keywords: algebra, equivalent expressions, simplifying algebraic expressions, combining like terms, algebra practice, mathematical skills, expression comparison, algebraic manipulation
