Question:

Step 1: Set up the first equation by substituting $x = 7$ into the functional equation. We substitute $x = 7$ into the given functional equation $3f(x) + 2f\left(\frac{x+59}{x-1}\right) = 10x + 30$. First, calculate the argument of the second function: $$\frac{7+59}{7-1} = \frac{66}{6} = 11$$ Substituting into the functional equation: $$3f(7) + 2f(11) = 10(7) + 30 = 100 \quad \text{...(1)}$$ Step 2: Set up the second equation by substituting $x = 11$ into the functional equation. We substitute $x = 11$ into the functional equation. First, calculate the argument of the second function: $$\frac{11+59}{11-1} = \frac{70}{10} = 7$$ Substituting into the functional equation: $$3f(11) + 2f(7) = 10(11) + 30 = 140 \quad \text{...(2)}$$ Step 3: Express $f(11)$ in terms of $f(7)$ using equation (1). From equation (1): $$3f(7) + 2f(11) = 100$$ $$2f(11) = 100 - 3f(7)$$ $$f(11) = 50 - 1.5f(7)$$ Step 4: Substitute the expression for $f(11)$ into equation (2). Substituting $f(11) = 50 - 1.5f(7)$ into equation (2): $$3\left(50 - 1.5f(7)\right) + 2f(7) = 140$$ $$150 - 4.5f(7) + 2f(7) = 140$$ $$150 - 2.5f(7) = 140$$ Step 5: Solve for $f(7)$. $$-2.5f(7) = 140 - 150$$ $$-2.5f(7) = -10$$ $$f(7) = \frac{-10}{-2.5} = 4$$ **Final Answer:** The value of $f(7) = 4$, which corresponds to **Option 2**.