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.
