The number of integer oalues of m, for which the x-coordinaie of the point of intersection of the lines $3x + 4y = 9$and $y = mx + 1$ is also an integer

The coordinates of the point of intersection of the given lines are

$\left(\frac{5}{3+4m}-\frac{3+9m}{3+4m}\right)$

It is given that the x-coordinate of this point is an integer

$\therefore \frac{5}{3+4m}\text{ is an integer }$

$\begin{array}{l}\Rightarrow 3+4m\text{ is a divisor of }5\\ \Rightarrow 3+4m=1,3+4m=-1,3+4m=5,3+4m=-5\\ \Rightarrow m=-1,-2\left[\begin{array}{c}\because 3+4m=1,3+4m=5\text{ do not }\\ \text{ give integral values of }m\end{array}\right]\end{array}$

Hence, there are two values of $m$.