Simplify, using the associative property of multiplication.
$ = 48x - 30y - \square$

The missing term is 6 ( 8 x − 5 y ) . Using the associative property of multiplication, we can simplify the expression to 0. This illustrates how regrouping terms can lead to simplification in algebraic expressions.
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Answered by Anonymous | 2025-07-04

Assume the missing term is 6 ( 8 x − 5 y ) .
Apply the associative property: 6 ( 8 x − 5 y ) = 48 x − 30 y .
Substitute the missing term back into the original expression: 48 x − 30 y − ( 48 x − 30 y ) .
Simplify the expression: 48 x − 30 y − 48 x + 30 y = 0 . The missing term is 6 ( 8 x − 5 y ) ​ .

Explanation

Understanding the Problem We are given the expression 48 x − 30 y − □ and asked to simplify it using the associative property of multiplication. The goal is to determine the missing term represented by □ .

Simplifying the Missing Term Let's assume the missing term is 6 ( 2 x + 3 y − 5 x + 8 y ) . We need to simplify this expression using the associative property. First, we simplify the terms inside the parenthesis: 2 x − 5 x = − 3 x and 3 y + 8 y = 11 y . So the expression inside the parenthesis becomes − 3 x + 11 y . Now we multiply the entire expression by 6: 6 ( − 3 x + 11 y ) = 6 × − 3 x + 6 × 11 y = − 18 x + 66 y .

Combining Like Terms Now, we substitute the simplified missing term back into the original expression: 48 x − 30 y − ( − 18 x + 66 y ) = 48 x − 30 y + 18 x − 66 y . Combining like terms, we have ( 48 x + 18 x ) + ( − 30 y − 66 y ) = 66 x − 96 y .

Finding the Missing Term Therefore, if the missing term is 6 ( 2 x + 3 y − 5 x + 8 y ) , the simplified expression is 66 x − 96 y . However, the problem asks us to find the missing term, not simplify the entire expression. Let's assume the missing term is 6 ( 8 x − 5 y ) . Then, using the associative property, we have 6 ( 8 x − 5 y ) = 6 × 8 x − 6 × 5 y = 48 x − 30 y . Substituting this back into the original expression, we get 48 x − 30 y − ( 48 x − 30 y ) = 48 x − 30 y − 48 x + 30 y = 0 .

Trying Another Approach Let's try another approach. Suppose the missing term is 5 ( 3 y + 4 x + x − 10 y ) . Simplifying inside the parentheses, we get 4 x + x = 5 x and 3 y − 10 y = − 7 y . So the expression inside the parentheses is 5 x − 7 y . Multiplying by 5, we have 5 ( 5 x − 7 y ) = 25 x − 35 y . Substituting this back into the original expression, we get 48 x − 30 y − ( 25 x − 35 y ) = 48 x − 30 y − 25 x + 35 y = ( 48 x − 25 x ) + ( − 30 y + 35 y ) = 23 x + 5 y .

Final Answer Let's assume the missing term is 24 x . Then the expression becomes 48 x − 30 y − 24 x = 48 x − 24 x − 30 y = 24 x − 30 y . This doesn't seem to lead to a simplification using the associative property.


Let's assume the missing term is 6 ( 3 x + 5 y ) . Then 6 ( 3 x + 5 y ) = 18 x + 30 y . So the expression becomes 48 x − 30 y − ( 18 x + 30 y ) = 48 x − 30 y − 18 x − 30 y = 30 x − 60 y . This also doesn't seem to be the correct approach.
Let's assume the missing term is 6 ( 8 x − 5 y ) . Then the expression becomes 48 x − 30 y − 6 ( 8 x − 5 y ) = 48 x − 30 y − ( 48 x − 30 y ) = 48 x − 30 y − 48 x + 30 y = 0 . So the missing term is 6 ( 8 x − 5 y ) .

Conclusion The missing term is 6 ( 8 x − 5 y ) .

Examples
Associative property is used in many real life scenarios like calculating the total cost of items with discounts. For example, if you buy 3 items each costing $10 and get a 20% discount on each item, you can calculate the total cost as 3 * (10 - 10*0.20) = 3 * (10 - 2) = 3 * 8 = $24. The associative property helps simplify such calculations.

Answered by GinnyAnswer | 2025-07-04