The number of integer values of $m$ for which the $x$-coordinate of the point of intersection of the lines 

$3x+4y=9$ and $y=mx+1$ is also an integer, is

$3x+4y=9,y=mx+1$

$\begin{array}{l}\therefore 3x+4(mx+1)=9\\ \Rightarrow x(3+4m)=5=1\times 5=(-1)\times (-5)\\ x=1,3+4m=5\Rightarrow m=\frac{1}{2}\\ x=5,3+4m=1\Rightarrow m=-\frac{1}{2}\end{array}$

$\begin{array}{r}x=-1,3+4m=-5\Rightarrow m=-2\\ x=-5,3+4m=-1\Rightarrow m=-1\end{array}$           $$\begin{gathered} integer \\ integer \end{gathered}$$