Which expression is equivalent to $(3 a+4 a b-7 b)-(a+2 a b-4 b)$?
Answer (2)
Distribute the negative sign: ( 3 a + 4 ab − 7 b ) − ( a + 2 ab − 4 b ) = 3 a + 4 ab − 7 b − a − 2 ab + 4 b .
Combine like terms: 3 a − a + 4 ab − 2 ab − 7 b + 4 b .
Simplify the coefficients: ( 3 − 1 ) a + ( 4 − 2 ) ab + ( − 7 + 4 ) b = 2 a + 2 ab − 3 b .
The equivalent expression is 2 a + 2 ab − 3 b .
Explanation
Understanding the Problem We are given the expression ( 3 a + 4 ab − 7 b ) − ( a + 2 ab − 4 b ) . Our goal is to simplify this expression by combining like terms.
Distributing the Negative Sign First, distribute the negative sign to each term inside the second parentheses: ( 3 a + 4 ab − 7 b ) − ( a + 2 ab − 4 b ) = 3 a + 4 ab − 7 b − a − 2 ab + 4 b
Combining Like Terms Next, we combine the like terms. We have terms with a , terms with ab , and terms with b . So, we group them together: 3 a − a + 4 ab − 2 ab − 7 b + 4 b
Simplifying the Coefficients Now, we simplify each group by performing the arithmetic: ( 3 − 1 ) a + ( 4 − 2 ) ab + ( − 7 + 4 ) b = 2 a + 2 ab − 3 b
Final Answer Therefore, the simplified expression is 2 a + 2 ab − 3 b .
Examples
This type of algebraic simplification is useful in many real-world scenarios. For example, imagine you are managing inventory for a store. You might have an expression representing the number of items you have in stock, and another expression representing the number of items that were sold or returned. Simplifying the combined expression helps you quickly determine your current inventory level. This is also applicable in calculating profits, where you subtract costs from revenue to find the net profit.
The simplified expression for ( 3 a + 4 ab − 7 b ) − ( a + 2 ab − 4 b ) is 2 a + 2 ab − 3 b . First, distribute the negative sign, then combine like terms to find the final answer. Therefore, the answer is 2 a + 2 ab − 3 b .
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