Let $f$ be a function such that $f(x) + 3\,f\bigl(24/x\bigr) = 4x$, $x\neq0$. Then $f(3) + f(8)$ is equal to:

Step 1: Set up equations for f(3) and f(8)

We are given a functional equation. To find $f(3)$ and $f(8)$, we will substitute $x=3$ and $x=8$ into the given equation to form a system of two equations.

Step 2: Substitute $x = 3$ and $x = 8$

First, substitute $x=3$ into the original equation. This gives us $f(3) + 3f(8) = 12$. Next, substitute $x=8$ into the original equation. This gives us $f(8) + 3f(3) = 32$. Now we have a system of two linear equations with two unknowns, $f(3)$ and $f(8)$.

Step 3: Solve the system of equations

We have the system of equations: $f(3) + 3f(8) = 12$ and $3f(3) + f(8) = 32$. We can solve this system by multiplying the first equation by 3 and subtracting the second equation from it, or by adding the two equations directly.

Step 4: Add the two equations

Adding equation (1) and equation (2) directly simplifies the problem. We get $4f(3) + 4f(8) = 44$. Factoring out 4, we have $4(f(3) + f(8)) = 44$.

Step 5: Find f(3) + f(8)

Dividing both sides of the equation $4(f(3) + f(8)) = 44$ by 4, we find that $f(3) + f(8) = 11$. This directly gives us the required sum.

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