What must be added to [tex]$60(5 x+4 y)$[/tex] to get [tex]$(2 x-10 y)$[/tex]?
Answer (2)
Set up the equation: 60 ( 5 x + 4 y ) + A = 2 x − 10 y .
Isolate A : A = ( 2 x − 10 y ) − 60 ( 5 x + 4 y ) .
Distribute and simplify: A = 2 x − 10 y − 300 x − 240 y .
Combine like terms: A = − 298 x − 250 y . The expression to be added is − 298 x − 250 y .
Explanation
Problem Setup We are given an expression 60 ( 5 x + 4 y ) and we want to find an expression that, when added to it, results in ( 2 x − 10 y ) . Let the expression to be added be denoted by A . Then, we can write the equation: 60 ( 5 x + 4 y ) + A = 2 x − 10 y
Isolating the Unknown Our goal is to isolate A on one side of the equation. To do this, we subtract 60 ( 5 x + 4 y ) from both sides: A = ( 2 x − 10 y ) − 60 ( 5 x + 4 y )
Distributing Now, we need to simplify the expression on the right side. First, distribute the 60 across the terms inside the parentheses: A = 2 x − 10 y − ( 300 x + 240 y )
Combining Like Terms Next, remove the parentheses and combine like terms: A = 2 x − 10 y − 300 x − 240 y A = ( 2 x − 300 x ) + ( − 10 y − 240 y )
Final Simplification Finally, simplify the expression by performing the subtractions: A = − 298 x − 250 y So, the expression that must be added to 60 ( 5 x + 4 y ) to get ( 2 x − 10 y ) is − 298 x − 250 y .
Examples
Imagine you're adjusting a recipe. You have a large batch of cookies, represented by 60 ( 5 x + 4 y ) , but you want a smaller batch, represented by ( 2 x − 10 y ) . The expression − 298 x − 250 y tells you exactly how much to remove from the original batch to achieve your desired quantity. This kind of algebraic manipulation is useful in many real-life scenarios, such as adjusting quantities in recipes, managing budgets, or altering resource allocations to meet specific goals.
To determine what must be added to 60 ( 5 x + 4 y ) to equal ( 2 x − 10 y ) , we have to find A such that A = − 298 x − 250 y . Therefore, adding − 298 x − 250 y will balance the equation. This means that we subtract these amounts to adjust the first expression to reach the second expression.
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