Which expression is equivalent to [tex]$6 x^3+3 y^2-5 x^3+2 y^2$[/tex]?
A) [tex]$x^6+3 y^4$[/tex]
B) [tex]$6 x^6 y^4$[/tex]
C) [tex]$6 x^3 y^2$[/tex]
D) [tex]$x^3+5 y^2$[/tex]
Answer (1)
Combine like terms: Group x 3 and y 2 terms.
Simplify x 3 terms: 6 x 3 − 5 x 3 = x 3 .
Simplify y 2 terms: 3 y 2 + 2 y 2 = 5 y 2 .
The equivalent expression is x 3 + 5 y 2 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to 6 x 3 + 3 y 2 − 5 x 3 + 2 y 2 . This involves combining like terms.
Grouping Like Terms First, let's group the like terms together: 6 x 3 − 5 x 3 + 3 y 2 + 2 y 2
Combining x^3 Terms Now, combine the x 3 terms: ( 6 − 5 ) x 3 = 1 x 3 = x 3
Combining y^2 Terms Next, combine the y 2 terms: ( 3 + 2 ) y 2 = 5 y 2
Simplified Expression So, the simplified expression is: x 3 + 5 y 2
Final Answer Comparing this with the given choices, we see that it matches option D.
Examples
Imagine you're organizing your bookshelf. You have different types of books: some are science fiction (represented by x 3 ) and others are fantasy (represented by y 2 ). If you start with 6 science fiction books and then remove 5, you're left with 1 science fiction book. Similarly, if you have 3 fantasy books and acquire 2 more, you end up with 5 fantasy books. Combining these, you have 1 science fiction book and 5 fantasy books, which is represented by the expression x 3 + 5 y 2 . This problem demonstrates how combining like terms can simplify real-world scenarios involving different categories or types of items.
