Answered

A local museum charges [tex]$25 per adult and $[/tex]12 per child for admission fees. At the end of the day, the museum made $9,014 in total admission revenue and had a total of 450 guests.

The system of equations below models the number of guests that were children, [tex]\( x \)[/tex], and the number of guests that were adults, [tex]\( y \)[/tex]:
[tex]\[
\begin{aligned}
12x + 25y & = 9,014 \\
x + y & = 450
\end{aligned}
\][/tex]

1. Place a point on the graph representing the solution to the system of equations. One of the equations has already been graphed.
2. Determine the approximate number of guests that were children and the number of guests that were adults for that day.

Possible answers:
- 275
- 450
- 350
- 200
- 300
- 175

Answer :

GinnyAnswer
To determine the number of children and adults at the museum, we need to solve the given system of equations:

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

Here's the step-by-step solution to this problem:

1. Express one variable in terms of the other using the simpler equation (second equation).

From the equation [tex]\( x + y = 450 \)[/tex], we can solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:

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

2. Substitute this expression into the first equation.

Substitute [tex]\( y = 450 - x \)[/tex] into the equation [tex]\( 12x + 25y = 9014 \)[/tex]:

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

3. Simplify and solve for [tex]\( x \)[/tex].

Distribute and combine like terms:

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

Simplify the equation:

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

Isolate [tex]\( x \)[/tex] by subtracting 11250 from both sides:

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

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

Divide both sides by -13 to solve for [tex]\( x \)[/tex]:

[tex]\[ x = \frac{2236}{13} \approx 172 \][/tex]

4. Find [tex]\( y \)[/tex] using the value of [tex]\( x \)[/tex].

Substitute [tex]\( x = 172 \)[/tex] back into the equation [tex]\( y = 450 - x \)[/tex]:

[tex]\[ y = 450 - 172 = 278 \][/tex]

Therefore, the number of children ([tex]\(x\)[/tex]) that visited the museum is approximately [tex]\(172\)[/tex], and the number of adults ([tex]\(y\)[/tex]) is approximately [tex]\(278\)[/tex].

5. Verify the result using the choices provided.

The choices given are: [tex]\(275\)[/tex], [tex]\(450\)[/tex], [tex]\(350\)[/tex], [tex]\(200\)[/tex], [tex]\(300\)[/tex], [tex]\(175\)[/tex]. We can confirm that none of these choices exactly fit [tex]\(172\)[/tex] for children or [tex]\(278\)[/tex] for adults, meaning the solution ([tex]\(172\)[/tex] children and [tex]\(278\)[/tex] adults) is neither overestimated nor underestimated based on the given constraints.

In conclusion, approximately [tex]\(172\)[/tex] children and [tex]\(278\)[/tex] adults visited the museum.