To determine which pair of expressions are equivalent, we need to simplify both expressions within each pair and then compare them. Let's analyze each option:
Option A:
Expressions: [tex]\(2y - 4\)[/tex] and [tex]\(2y - 4 + 9\)[/tex]
Simplifying the second expression:
[tex]\[ 2y - 4 + 9 = 2y + 5 \][/tex]
Comparing with the first expression:
[tex]\[ 2y - 4 \neq 2y + 5 \][/tex]
Therefore, [tex]\(2y - 4\)[/tex] and [tex]\(2y - 4 + 9\)[/tex] are not equivalent.
Option B:
Expressions: [tex]\(14x - 7\)[/tex] and [tex]\(7x - 14\)[/tex]
These are already simplified, but let’s compare them directly:
[tex]\[ 14x - 7 \neq 7x - 14 \][/tex]
Therefore, [tex]\(14x - 7\)[/tex] and [tex]\(7x - 14\)[/tex] are not equivalent.
Option C:
Expressions: [tex]\(9y + 3 + 2\)[/tex] and [tex]\(3(y + 1) + 2\)[/tex]
Simplifying both expressions:
[tex]\[ 9y + 3 + 2 = 9y + 5 \][/tex]
[tex]\[ 3(y + 1) + 2 = 3y + 3 + 2 = 3y + 5 \][/tex]
Comparing them:
[tex]\[ 9y + 5 = 3y + 5 \][/tex]
Therefore, [tex]\(9y + 3 + 2\)[/tex] and [tex]\(3(y + 1) + 2\)[/tex] are equivalent.
Option D:
Expressions: [tex]\(x + y\)[/tex] and [tex]\(x - y + 2y\)[/tex]
Simplifying the second expression:
[tex]\[ x - y + 2y = x + y \][/tex]
Comparing them:
[tex]\[ x + y = x + y \][/tex]
Therefore, [tex]\(x + y\)[/tex] and [tex]\(x - y + 2y\)[/tex] are equivalent.
Option E:
Expressions: [tex]\(x + 2y\)[/tex] and [tex]\(x - y\)[/tex]
These are already simplified, so let’s compare them directly:
[tex]\[ x + 2y \neq x - y \][/tex]
Therefore, [tex]\(x + 2y\)[/tex] and [tex]\(x - y\)[/tex] are not equivalent.
Conclusion:
After examining all the options, the equivalent pairs of expressions are found in Option C and Option D. Therefore, the correct answer includes the pairs provided in these options:
Answer: C and D
