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Given the domain [tex]$\{-4,0,5\}$[/tex], what is the range for the relation [tex]$12x + 6y = 24$[/tex]?

A. [tex]$\{-4,4,14\}$[/tex]
B. [tex]$\{-12,-4,6\}$[/tex]
C. [tex]$\{12,4,-6\}$[/tex]
D. [tex]$\{2,4,9\}$[/tex]

Sagot :

GinnyAnswer
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]
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