To determine the number of hats [tex]\(x\)[/tex] and T-shirts [tex]\(y\)[/tex] Pablo can buy within his constraints, we need to solve the given system of linear equations:
[tex]\[
\begin{cases}
5x + 10y = 400 \\
x + y = 50
\end{cases}
\][/tex]
Let's solve it step by step:
1. First, we'll simplify the second equation to solve for one of the variables. Let's solve for [tex]\(y\)[/tex]:
[tex]\[
x + y = 50 \implies y = 50 - x
\][/tex]
2. Next, we'll substitute [tex]\(y = 50 - x\)[/tex] into the first equation to eliminate [tex]\(y\)[/tex]:
[tex]\[
5x + 10(50 - x) = 400
\][/tex]
3. Distribute the 10:
[tex]\[
5x + 500 - 10x = 400
\][/tex]
4. Combine like terms:
[tex]\[
-5x + 500 = 400
\][/tex]
5. Isolate [tex]\(x\)[/tex]:
[tex]\[
-5x = 400 - 500
\][/tex]
[tex]\[
-5x = -100
\][/tex]
[tex]\[
x = \frac{-100}{-5}
\][/tex]
[tex]\[
x = 20
\][/tex]
6. Now we have [tex]\(x = 20\)[/tex]. Substitute this back into [tex]\(y = 50 - x\)[/tex] to find [tex]\(y\)[/tex]:
[tex]\[
y = 50 - 20
\][/tex]
[tex]\[
y = 30
\][/tex]
Therefore, the solution to the system is [tex]\(x = 20\)[/tex] and [tex]\(y = 30\)[/tex]. The correct ordered pair that satisfies both equations is:
[tex]\[
(20, 30)
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{(20, 30)}
\][/tex]
