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In Mathematics / College | 2025-07-03

Select the correct answer.

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

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

Asked by jonatan65127

Answer (2)

Solve the equation for y in terms of x : y = 4 − 2 x .
Substitute x = − 4 into the equation: y = 4 − 2 ( − 4 ) = 12 .
Substitute x = 0 into the equation: y = 4 − 2 ( 0 ) = 4 .
Substitute x = 5 into the equation: y = 4 − 2 ( 5 ) = − 6 . The range is { 12 , 4 , − 6 } ​ .

Explanation

Understanding the Problem We are given the relation 12 x + 6 y = 24 and the domain { − 4 , 0 , 5 } . Our goal is to find the range of this relation. The range is the set of all possible y values that we get when we plug in the x values from the domain into the equation.

Solving for y First, let's solve the equation 12 x + 6 y = 24 for y in terms of x . We can do this by subtracting 12 x from both sides of the equation: 6 y = 24 − 12 x
Now, divide both sides by 6: y = 6 24 − 12 x ​ = 4 − 2 x

Calculating the Range Now, we will substitute each value from the domain { − 4 , 0 , 5 } into the equation y = 4 − 2 x to find the corresponding y values.


For x = − 4 :
y = 4 − 2 ( − 4 ) = 4 + 8 = 12
For x = 0 :
y = 4 − 2 ( 0 ) = 4 − 0 = 4
For x = 5 :
y = 4 − 2 ( 5 ) = 4 − 10 = − 6

Determining the Range The set of y values we obtained is { 12 , 4 , − 6 } . Therefore, the range of the relation is { 12 , 4 , − 6 } .

Final Answer The range for the relation 12 x + 6 y = 24 with the given domain { − 4 , 0 , 5 } is { 12 , 4 , − 6 } . So, the correct answer is A.


Examples
In real life, this type of problem can be used to model relationships between two variables where one variable depends on the other. For example, suppose you have a budget of $24 to spend on two items, where one item costs $12 per unit and the other costs $6 per unit. If you can only buy -4, 0, or 5 units of the first item, this problem helps you determine how many units of the second item you can buy to stay within your budget. Understanding such relationships is crucial in economics, finance, and resource allocation.

Answered by GinnyAnswer | 2025-07-03

To find the range of the relation 12 x + 6 y = 24 with the domain {-4, 0, 5}, we rearrange to get y = 4 − 2 x . Substituting the values from the domain gives us the range {12, 4, -6}. Thus, the correct answer is A: {12, 4, -6}.
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Answered by Anonymous | 2025-07-04

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