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?

If we let $x=0$ then we do indeed find that the expression equals $21-9y$. But this does not mean that $21-9y$ must equal $0$; it only means that $21-9y$ is a constant. If, as Raffaele has suggested, we let $x=1$, then we find that the expression equals $24-11y$. Hence, $21-9y$ and $24-11y$ must be equal to the same constant, and so solving the problem boils down to solving the equation $21-9y=24-11y$:
\begin{align}
21 - 9y &= 24 - 11y &&\text{Subtract $21$ from both sides}\\
-9y &= 3 - 11y &&\text{Add $11y$ to both sides}\\
2y &= 3 &&\text{Divide both sides by $2$}\\
y &= \frac{3}{2} \, .
\end{align}
