If $3(x +7) - y(2x+9)$ is the same for all values of $x$, what number must $y$ be? If we let $x = 0$.
\begin{align*} 
3(0+7)-y(2(0)+9) \\
21-9y \\
\end{align*}
Then $9y$ should always equal $21$?
Solving for $y$ finds $\frac{7}{3}$.
But $3(x+7)-\frac{7}{3}(2(x)+9)$ does not have the same result for diffrent values of $x$.
Where am I going wrong?

You have the function:
$$ f \left( x, y \right) = 3 \left( x + 7 \right) - y \left( 2 x + 9 \right) $$
Since the function is constant for $ x $ then it means:
$$ \nabla_{x} f \left( x, y \right) = 0 = 3 - 2 y \Rightarrow y = \frac{3}{2} $$
Another way thinking about it, is by definition the function with respect to $ x $ must be constant (Specific case of Linear).
Hence its coefficients (Which are given by the derivative since it's a specific case of Linear Function) must vanish (As seen in @Deepak answer).
