Verified Answer :
- Simplify the first expression \(8a^2 - 4b + 7a^2\) by combining the like terms \(8a^2\) and \(7a^2\), resulting in \(15a^2 - 4b\).
- Simplify the second expression \(15a^2 - 10b + 6b\) by combining the like terms \(-10b\) and \(6b\), resulting in \(15a^2 - 4b\).
- Compare the two simplified expressions and conclude that they are identical.
- Final Answer: \(\boxed{\text{Yes}}\)
Explanation
- Understanding Algebraic Equivalence
To determine if two algebraic expressions are equivalent, we need to check if they have the same value for all possible values of the variables. This is most easily done by simplifying both expressions by combining like terms. Like terms are terms that have the same variables raised to the same powers. In our case, the terms involved are \(a^2\) and \(b\).
- Simplifying the First Expression
We first simplify the expression \(8a^2 - 4b + 7a^2\).
To do this, we group the terms that contain \(a^2\):
\[8a^2 + 7a^2 - 4b\]
Adding the coefficients of \(a^2\) gives \(8 + 7 = 15\). Thus, the expression becomes:
\[15a^2 - 4b\]
- Simplifying the Second Expression
Next, we simplify the expression \(15a^2 - 10b + 6b\).
Here, we group the terms that contain \(b\):
\[15a^2 + (-10b + 6b)\]
Combining the coefficients of \(b\), we have \(-10 + 6 = -4\). Thus, the expression becomes:
\[15a^2 - 4b\]
- Comparing and Concluding
Comparing the simplified forms of both expressions, we see that both simplify to exactly \(15a^2 - 4b\). Since both expressions reduce to the same polynomial, they are equivalent for all values of \(a\) and \(b\). Therefore, the answer is yes, they are equivalent. \(\boxed{\text{Yes}}\)
Examples
Algebraic equivalence is a fundamental concept in mathematics that helps in simplifying complex equations and verifying solutions. For instance, if you are designing a budget for a project and calculate costs using two different formulas—such as total cost as 5 units of labor + 3 units of materials versus 2 units of labor + 3 units of materials + 3 units of labor—understanding that these expressions are equivalent confirms that your accounting is consistent regardless of how you group the items. This ability to manipulate expressions ensures that calculations remain accurate during complex problem-solving.