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A local museum charges [tex]$\$[/tex] 25[tex]$ per adult and $[/tex]\[tex]$ 12$[/tex] per child for admission fees. At the end of a day, the museum made [tex]$\$[/tex] 9,014[tex]$ in total admission revenue, not including sales tax, and had a total of 450 guests.

The system of equations below can be used to model the number of guests that were children, $[/tex]x[tex]$, and the number of guests that were adults, $[/tex]y$.

[tex]\[
\begin{aligned}
12x + 25y &= 9,014 \\
x + y &= 450
\end{aligned}
\][/tex]

Graph the equations and determine the approximate number of guests that were children and the approximate number of guests that were adults for that day.

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275
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450

Sagot :

GinnyAnsweridn
To solve for the number of guests that were children (denoted as [tex]\( x \)[/tex]) and the number of guests that were adults (denoted as [tex]\( y \)[/tex]), we analyze the given system of equations:

[tex]\[ \begin{aligned} 1. \quad 12x + 25y &= 9014 \\ 2. \quad x + y &= 450 \end{aligned} \][/tex]

First, we solve the second equation for one of the variables. Let's solve for [tex]\( y \)[/tex]:

[tex]\[ y = 450 - x \][/tex]

Next, we substitute [tex]\( y = 450 - x \)[/tex] into the first equation:

[tex]\[ 12x + 25(450 - x) = 9014 \][/tex]

Expanding and simplifying the equation:

[tex]\[ 12x + 11250 - 25x = 9014 \][/tex]

Combine like terms:

[tex]\[ -13x + 11250 = 9014 \][/tex]

Subtract 11250 from both sides of the equation:

[tex]\[ -13x = 9014 - 11250 \][/tex]

[tex]\[ -13x = -2236 \][/tex]

Divide both sides by -13:

[tex]\[ x = \frac{-2236}{-13} \][/tex]

[tex]\[ x = 172 \][/tex]

So, the number of children is [tex]\( 172 \)[/tex].

To find the number of adults, we substitute [tex]\( x = 172 \)[/tex] back into the second equation:

[tex]\[ x + y = 450 \][/tex]

[tex]\[ 172 + y = 450 \][/tex]

Subtract 172 from both sides:

[tex]\[ y = 450 - 172 \][/tex]

[tex]\[ y = 278 \][/tex]

So, the number of adults is [tex]\( 278 \)[/tex].

Therefore, the approximate number of guests that were children for that day is 172, and the approximate number of guests that were adults for that day is 278.
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