Which expressions are equivalent when [tex]$y=2$[/tex] and [tex]$y=5$[/tex]?
[tex]$2 y-1$[/tex] and [tex]$3 y-5+y$[/tex]
[tex]$5 y+4$[/tex] and [tex]$7 y+4-2 y$[/tex]
[tex]$y+7$[/tex] and [tex]$y+2+y$[/tex]
[tex]$3 y-4$[/tex] and [tex]$3 y-2+y$[/tex]
Answer (2)
Substitute y = 2 and y = 5 into each pair of expressions.
Simplify each expression after substitution.
Compare the values of the expressions in each pair for both y = 2 and y = 5 .
Only Pair 2, 5 y + 4 and 7 y + 4 − 2 y , yields equal values for both y = 2 and y = 5 , thus they are equivalent. 5 y + 4 and 7 y + 4 − 2 y
Explanation
Problem Analysis We are given four pairs of expressions and we need to find out which pairs are equivalent when y = 2 and y = 5 . To do this, we will substitute y = 2 and y = 5 into each expression and check if the values are equal.
Pair 1 Evaluation Pair 1: 2 y − 1 and 3 y − 5 + y When y = 2 :
2 ( 2 ) − 1 = 4 − 1 = 3 3 ( 2 ) − 5 + 2 = 6 − 5 + 2 = 3 When y = 5 :
2 ( 5 ) − 1 = 10 − 1 = 9 3 ( 5 ) − 5 + 5 = 15 − 5 + 5 = 15 Since the values are equal when y = 2 but not when y = 5 , this pair is not equivalent.
Pair 2 Evaluation Pair 2: 5 y + 4 and 7 y + 4 − 2 y When y = 2 :
5 ( 2 ) + 4 = 10 + 4 = 14 7 ( 2 ) + 4 − 2 ( 2 ) = 14 + 4 − 4 = 14 When y = 5 :
5 ( 5 ) + 4 = 25 + 4 = 29 7 ( 5 ) + 4 − 2 ( 5 ) = 35 + 4 − 10 = 29 Since the values are equal for both y = 2 and y = 5 , this pair is equivalent.
Pair 3 Evaluation Pair 3: y + 7 and y + 2 + y When y = 2 :
2 + 7 = 9 2 + 2 + 2 = 6 When y = 5 :
5 + 7 = 12 5 + 2 + 5 = 12 Since the values are not equal when y = 2 , this pair is not equivalent.
Pair 4 Evaluation Pair 4: 3 y − 4 and 3 y − 2 + y When y = 2 :
3 ( 2 ) − 4 = 6 − 4 = 2 3 ( 2 ) − 2 + 2 = 6 − 2 + 2 = 6 When y = 5 :
3 ( 5 ) − 4 = 15 − 4 = 11 3 ( 5 ) − 2 + 5 = 15 − 2 + 5 = 18 Since the values are not equal when y = 2 and y = 5 , this pair is not equivalent.
Conclusion Therefore, only the expressions in Pair 2 are equivalent when y = 2 and y = 5 .
Examples
Understanding equivalent expressions is crucial in algebra, especially when simplifying equations or solving for unknowns. For instance, imagine you're designing a rectangular garden where the length can be expressed in two different ways depending on available materials. If both expressions yield the same length for specific widths, you know your design is consistent, ensuring the garden fits perfectly in your yard. This concept extends to various fields like engineering, economics, and computer science, where different formulas might represent the same underlying principle or quantity.
The expressions in Pair 2, 5 y + 4 and 7 y + 4 − 2 y , are equivalent when y = 2 and y = 5 as they yield the same values for both substitutions. Other pairs do not maintain equal values for both inputs. Therefore, the answer is confirmed as Pair 2 is the only equivalent pair.
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