To determine which expressions are equivalent to [tex]\( j + j + 2k \)[/tex], let's analyze each option:
1. Given expression: [tex]\( j + j + 2k \)[/tex]
- Combine like terms: [tex]\( j + j = 2j \)[/tex]
- Therefore, the given expression simplifies to [tex]\( 2j + 2k \)[/tex].
Now, let's consider each option:
Option (A): [tex]\( 2jk \)[/tex]
- This expression involves the product of [tex]\( j \)[/tex] and [tex]\( k \)[/tex] multiplied by 2.
- There is no sum of [tex]\( j \)[/tex] and [tex]\( k \)[/tex] involved, and it is fundamentally different from the linear terms in the original expression.
- Therefore, [tex]\( 2jk \)[/tex] is not equivalent to [tex]\( 2j + 2k \)[/tex].
Option (B): [tex]\( 2(j + j + k) \)[/tex]
- First, simplify inside the parentheses: [tex]\( j + j + k = 2j + k \)[/tex].
- Then distribute the 2: [tex]\( 2 \times (2j + k) = 4j + 2k \)[/tex].
- This results in [tex]\( 4j + 2k \)[/tex], which is not the same as the original expression [tex]\( 2j + 2k \)[/tex].
Option (C): None of the above
- Since neither option (A) nor option (B) is equivalent to the given expression [tex]\( j + j + 2k = 2j + 2k \)[/tex], this option remains valid.
In conclusion, neither [tex]\( 2jk \)[/tex] nor [tex]\( 2(j + j + k) \)[/tex] is equivalent to [tex]\( j + j + 2k \)[/tex].
Therefore, the correct answer is:
(c) None of the above
