To solve for (x) in the given system of equations, follow these steps:

Step 1: Eliminate fractions from the equations

First equation: (\frac{1}{2}x + \frac{1}{3}y = 16)
Multiply by 6 (LCM of 2 and 3):
(3x + 2y = 96) --- Equation A

Second equation: (\frac{1}{6}x + \frac{1}{5}y = 9)
Multiply by 30 (LCM of 6 and 5):
(5x + 6y = 270) --- Equation B

Step 2: Eliminate (y)

Multiply Equation A by 3:
(9x + 6y = 288) --- Equation C

Subtract Equation B from Equation C:
((9x + 6y) - (5x + 6y) = 288 - 270)
(4x = 18)

(x = \frac{18}{4} = \frac{9}{2} = 4.5)

Answer: (\boxed{4.5}) or (\boxed{\frac{9}{2}}). Since the problem likely expects an integer? Wait no, calculation confirms it's 4.5, but maybe as a fraction (\boxed{9/2}). However, if decimal is acceptable, (\boxed{4.5}). But in math problems, exact values are preferred, so (\boxed{9/2}) or (\boxed{4.5}). But let's check if the problem expects integer, maybe a typo? But according to given equations, it's 9/2.

Final Answer: (\boxed{4.5}) (or (\boxed{9/2}) if fraction is allowed). But perhaps the problem expects (\boxed{4.5}) as a decimal. However, in many cases, fractions are better, so (\boxed{9/2}) is correct.

(\boxed{9/2})

Wait wait, but let me see if the problem wants it as an integer. Maybe I made a mistake? Let's recheck: 3x+2y=96, 5x+6y=270. 3x=96-2y →x=(96-2y)/3. Plug into second equation:5(96-2y)/3 +6y=270 →(480 -10y +18y)/3=270 →480 +8y=810 →8y=330 →y=330/8=165/4=41.25, then x=(96-2(165/4))/3=(96 - 165/2)/3=(192 -165)/6=27/6=9/2=4.5, yes correct. So the answer is (\boxed{9/2}) or (\boxed{4.5}). Since the problem says "the value of x is__", probably expects a number, so (\boxed{4.5}) is okay, but 9/2 is exact. Maybe the problem expects 9/2, so I'll go with (\boxed{9/2}) but if it's decimal, 4.5. However, in most standard tests, fractions are preferred for exact answers, so (\boxed{9/2}).

(\boxed{9/2})

But wait, maybe the user expects integer, but no, the calculation is correct. So the answer is (\boxed{9/2}).

Final Answer: (\boxed{9/2})

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