Let's determine which pairs of expressions are equivalent for [tex]\( y = 2 \)[/tex] and [tex]\( y = 5 \)[/tex].
### Step-by-Step Analysis
1. First Pair: [tex]\( 2y - 1 \)[/tex] and [tex]\( 3y - 5 + y \)[/tex]
- For [tex]\( y = 2 \)[/tex]:
- [tex]\( 2y - 1 = 2(2) - 1 = 4 - 1 = 3 \)[/tex]
- [tex]\( 3y - 5 + y = 3(2) - 5 + 2 = 6 - 5 + 2 = 3 \)[/tex]
- For [tex]\( y = 5 \)[/tex]:
- [tex]\( 2y - 1 = 2(5) - 1 = 10 - 1 = 9 \)[/tex]
- [tex]\( 3y - 5 + y = 3(5) - 5 + 5 = 15 - 5 + 5 = 15 \)[/tex]
Since [tex]\( 2y - 1 = 3y - 5 + y \)[/tex] for [tex]\( y = 2 \)[/tex] but not for [tex]\( y = 5 \)[/tex], these expressions are not equivalent.
2. Second Pair: [tex]\( 5y + 4 \)[/tex] and [tex]\( 7y + 4 - 2y \)[/tex]
- For [tex]\( y = 2 \)[/tex]:
- [tex]\( 5y + 4 = 5(2) + 4 = 10 + 4 = 14 \)[/tex]
- [tex]\( 7y + 4 - 2y = 7(2) + 4 - 2(2) = 14 + 4 - 4 = 14 \)[/tex]
- For [tex]\( y = 5 \)[/tex]:
- [tex]\( 5y + 4 = 5(5) + 4 = 25 + 4 = 29 \)[/tex]
- [tex]\( 7y + 4 - 2y = 7(5) + 4 - 2(5) = 35 + 4 - 10 = 29 \)[/tex]
Since [tex]\( 5y + 4 = 7y + 4 - 2y \)[/tex] for both [tex]\( y = 2 \)[/tex] and [tex]\( y = 5 \)[/tex], these expressions are equivalent.
3. Third Pair: [tex]\( y + 7 \)[/tex] and [tex]\( y + 2 + y \)[/tex]
- For [tex]\( y = 2 \)[/tex]:
- [tex]\( y + 7 = 2 + 7 = 9 \)[/tex]
- [tex]\( y + 2 + y = 2 + 2 + 2 = 6 \)[/tex]
- For [tex]\( y = 5 \)[/tex]:
- [tex]\( y + 7 = 5 + 7 = 12 \)[/tex]
- [tex]\( y + 2 + y = 5 + 2 + 5 = 12 \)[/tex]
Since [tex]\( y + 7 \neq y + 2 + y \)[/tex] for [tex]\( y = 2 \)[/tex] but they equal for [tex]\( y = 5 \)[/tex], these expressions are not equivalent.
4. Fourth Pair: [tex]\( 3y - 4 \)[/tex] and [tex]\( 3y - 2 + y \)[/tex]
- For [tex]\( y = 2 \)[/tex]:
- [tex]\( 3y - 4 = 3(2) - 4 = 6 - 4 = 2 \)[/tex]
- [tex]\( 3y - 2 + y = 3(2) - 2 + 2 = 6 - 2 + 2 = 6 \)[/tex]
- For [tex]\( y = 5 \)[/tex]:
- [tex]\( 3y - 4 = 3(5) - 4 = 15 - 4 = 11 \)[/tex]
- [tex]\( 3y - 2 + y = 3(5) - 2 + 5 = 15 - 2 + 5 = 18 \)[/tex]
Since [tex]\( 3y - 4 \neq 3y - 2 + y \)[/tex] for both [tex]\( y = 2 \)[/tex] and [tex]\( y = 5 \)[/tex], these expressions are not equivalent.
### Conclusion
The only pair of expressions that is equivalent for both given values of [tex]\( y \)[/tex] is:
[tex]\[ 5y + 4 \quad \text{and} \quad 7y + 4 - 2y \][/tex]
