To solve for $f(8)$, let's analyze the functional equation given:

$2f(x) + 3f\left(\frac{2024}{x}\right) = 5x$

Step 1: Substitute $x = 8$

Let's first try substituting $x = 8$ directly to see if it gives us useful information.

$2f(8) + 3f\left(\frac{2024}{8}\right) = 5 \cdot 8$

Simplify the right side:

$2f(8) + 3f(253) = 40$

So we get one equation:

$2f(8) + 3f(253) = 40$

Step 2: Substitute $x = 253$

Now, let's try substituting $x = 253$ into the functional equation to get another relation.

$2f(253) + 3f\left(\frac{2024}{253}\right) = 5 \cdot 253$

Calculate $5 \cdot 253$:

$5 \cdot 253 = 1265$

So, we have:

$2f(253) + 3f(8) = 1265$

Step 3: Set up the system of equations

Now we have two equations:

1. $2f(8) + 3f(253) = 40$
2. $3f(8) + 2f(253) = 1265$

This is a system of linear equations in $f(8)$ and $f(253)$.

Step 4: Solve the system of equations

Let's solve for $f(8)$ and $f(253)$ by using the elimination or substitution method.

Multiply the first equation by 3 and the second equation by 2 to align the coefficients of $f(8)$:

$6f(8) + 9f(253) = 120$ $6f(8) + 4f(253) = 2530$

Now, subtract the first modified equation from the second:

$(6f(8) + 4f(253)) - (6f(8) + 9f(253)) = 2530 - 120$

$-5f(253) = 2410$

Divide by $-5$:

$f(253) = -482$

Step 5: Substitute back to find $f(8)$

Now substitute $f(253) = -482$ into the first equation:

$2f(8) + 3(-482) = 40$

$2f(8) - 1446 = 40$

Add 1446 to both sides:

$2f(8) = 1486$

Divide by 2:

$f(8) = 743$

Final Answer

$f(8) = 743$

Would you like further details on any steps or have any questions?

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1. How do functional equations generally work?
2. What techniques help in solving systems of equations like this one?
3. Are there alternative methods to solve functional equations?
4. How can we verify the solution in such problems?
5. Why did we choose $x = 8$ and $x = 253$ for substitution?

Tip: Solving functional equations often involves strategic substitutions to simplify the equation or reduce unknowns.