The function $f(x)$ satisfies
\[f(x) + f(x + 2y) = 6x + 6y - 8\]for all real numbers $x$ and $y.$  Find the value of $x$ such that $f(x) = 0.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
The function $f(x)$ satisfies
\[f(x) + f(x + 2y) = 6x + 6y - 8\]for all real numbers $x$ and $y.$  Find the value of $x$ such that $f(x) = 0.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{4}{3}
The function $f(x)$ satisfies
\[f(x) + f(x + 2y) = 6x + 6y - 8\]for all real numbers $x$ and $y.$  Find the value of $x$ such that $f(x) = 0.$
train