The number of intergral values of m, for which the x co-ordinate of the point of intersection of the lines $3x + 4y = 9,$ $y =mx + 1$ is an integer is 

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

$3x + 4(m x + 1) = 9$

$(3 + 4m)x = 5$

$x=\frac{5}{3+4m}$is an interger $\Rightarrow$$3+4m=\pm 1,\pm 5$

3 + 4m =1 $\Rightarrow$ m = $-\frac{1}{2}$  not an integer

3 + 4m =–1 $\Rightarrow$m = –1 integer

3 + 4m = 5 $\Rightarrow$m =$-\frac{1}{2}$ not an integer

3 + 4 m = –5  $\Rightarrow$ m = –2 integer

Number of integral values of m = 2