Let f be a function such that f(x)+3f(24/x)=4x, x=0. Then f(3) + f(8) is equal to:
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Step-by-Step Solution
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.