Answered

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


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

Verified Answer :

  • Rewrite the relation \(12x + 6y = 24\) as \(y = 4 - 2x\).
  • Substitute each value of the domain \(x \in \{-4, 0, 5\}\) into the equation to find the corresponding \(y\).
  • Calculate \(y = 12\) for \(x = -4\), \(y = 4\) for \(x = 0\), and \(y = -6\) for \(x = 5\).
  • The resulting range set is \(\{12, 4, -6\}\), which corresponds to the third choice: \(\boxed{\{12, 4, -6\}}\).

Explanation

  1. Analysis of the Problem and Data

The given relation is defined by the linear equation \(12x + 6y = 24\), and we are asked to find the range of the relation for the specific domain \(D = \{-4, 0, 5\}\). The domain is the set of allowed input values for \(x\), and the range is the set of all resulting output values for \(y\) corresponding to each input in the domain. We must solve for \(y\) in terms of \(x\) and then evaluate the expression for each element in the given domain.

  1. Solving for \(y\) in Terms of \(x\)

To express \(y\) in terms of \(x\), we rearrange the equation \(12x + 6y = 24\):
\[6y = 24 - 12x\]
Dividing the entire equation by 6, we get:
\[y = \frac{24}{6} - \frac{12x}{6}\]
\[y = 4 - 2x\]
This simplified linear equation allows us to calculate the corresponding \(y\) value for any given \(x\).

  1. Calculating the Range Values and Identifying the Final Answer

Now, substitute each element of the domain \(\{-4, 0, 5\}\) into the equation \(y = 4 - 2x\):

  • For \(x = -4\):

\[y = 4 - 2(-4) = 4 + 8 = 12\]

  • For \(x = 0\):

\[y = 4 - 2(0) = 4 - 0 = 4\]

  • For \(x = 5\):

\[y = 4 - 2(5) = 4 - 10 = -6\]
The resulting values for \(y\) are 12, 4, and -6. Thus, the range of the relation for the given domain is the set \(\{12, 4, -6\}\). Comparing this result with the given choices, we find it matches the third option.

Examples

Imagine you are budgeting for a party where the total cost is defined by a fixed amount and a variable cost based on guests. If the equation for your budget is \(12x + 6y = 24\) (where \(x\) represents the number of a certain item and \(y\) is the number of another), knowing the domain allows you to determine the possible values for \(y\) (the quantity of the second item) that keep your budget balanced. This technique is fundamental in resource management and optimization tasks in daily life.

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