To determine which expressions are equivalent to [tex]\(6xy + 9xy - 18y\)[/tex], we need to simplify and compare them.
Let's start simplification of the original expression:
[tex]\[ 6xy + 9xy - 18y \][/tex]
Combine like terms:
[tex]\[ (6xy + 9xy) - 18y = 15xy - 18y \][/tex]
Now, we will compare [tex]\(15xy - 18y\)[/tex] with each of the provided options:
A. [tex]\(3(2xy + 3xy - 6y)\)[/tex]
First, simplify within the parentheses:
[tex]\[ 2xy + 3xy - 6y = 5xy - 6y \][/tex]
Then, distribute the 3:
[tex]\[ 3(5xy - 6y) = 15xy - 18y \][/tex]
This matches [tex]\(15xy - 18y\)[/tex].
B. [tex]\(3x(2y + 3y - 6)\)[/tex]
First, simplify within the parentheses:
[tex]\[ 2y + 3y - 6 = 5y - 6 \][/tex]
Then, distribute the 3x:
[tex]\[ 3x(5y - 6) = 15xy - 18x \][/tex]
This expression does not match [tex]\(15xy - 18y\)[/tex].
C. [tex]\(3y(2x + 3 x - 6)\)[/tex]
First, simplify within the parentheses:
[tex]\[ 2x + 3x - 6 = 5x - 6 \][/tex]
Then, distribute the 3y:
[tex]\[ 3y(5x - 6) = 15xy - 18y \][/tex]
This matches [tex]\(15xy - 18y\)[/tex].
D. [tex]\(xy(5 - 6y)\)[/tex]
Distribute the [tex]\(xy\)[/tex]:
[tex]\[ xy(5 - 6y) = 5xy - 6xy^2 \][/tex]
This form does not match [tex]\(15xy - 18y\)[/tex].
E. [tex]\(3y(5x - 6)\)[/tex]
Distribute the 3y:
[tex]\[ 3y(5x - 6) = 15xy - 18y \][/tex]
This matches [tex]\(15xy - 18y\)[/tex].
F. [tex]\(y(15x - 18)\)[/tex]
Distribute the [tex]\(y\)[/tex]:
[tex]\[ y(15x - 18) = 15xy - 18y \][/tex]
This matches [tex]\(15xy - 18y\)[/tex].
Hence, the expressions equivalent to [tex]\(6xy + 9xy - 18y\)[/tex] are:
[tex]\[ \boxed{\text{A, C, E, and F}} \][/tex]
