Certainly! Let's solve for the range of the relation [tex]\( 12x + 6y = 24 \)[/tex] when the domain is given as [tex]\(\{-4, 0, 5\}\)[/tex].
We are given the equation:
[tex]\[ 12x + 6y = 24 \][/tex]
First, we solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ 6y = 24 - 12x \][/tex]
[tex]\[ y = 4 - 2x \][/tex]
Now, we substitute each value of [tex]\( x \)[/tex] from the domain [tex]\(\{-4, 0, 5\}\)[/tex] into the equation [tex]\( y = 4 - 2x \)[/tex] to find the corresponding values of [tex]\( y \)[/tex]:
1. For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = 4 - 2(-4) \][/tex]
[tex]\[ y = 4 + 8 \][/tex]
[tex]\[ y = 12 \][/tex]
2. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4 - 2(0) \][/tex]
[tex]\[ y = 4 \][/tex]
3. For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = 4 - 2(5) \][/tex]
[tex]\[ y = 4 - 10 \][/tex]
[tex]\[ y = -6 \][/tex]
So, the range corresponding to the domain [tex]\(\{-4, 0, 5\}\)[/tex] for the relation [tex]\( 12x + 6y = 24 \)[/tex] is:
[tex]\[ \{12, 4, -6\} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\{12, 4, -6\}} \][/tex]
