Question

The number of integral 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

• A. $2$
• B. $0$
• C. $4$
• D. $1$

Answer 1

The correct option is A

$2$

Explanation for the correct option

Given equations of lines are
$3x+4y=9...\left(1\right)$
and $y=mx+1\dots \left(2\right)$

Substituting $\left(2\right)$ in $\left(1\right)$
$\begin{array}{cc}& 3x+4\left(mx+1\right)=9\\ \Rightarrow & 3x+4mx+4=9\\ \Rightarrow & x\left(3+4m\right)=5\\ \Rightarrow & x=\frac{5}{3+4m}\end{array}$

For $x$ to be an integer, $3+4m$ must evenly divide $5$ i.e., $3+4m$ must be a factor of $5$

So,
$\begin{array}{ccccc}\Rightarrow & 3+4m=5& 3+4m=-5& 3+4m=1& 3+4m=-1\\ \Rightarrow & m=\frac{5-3}{4}& m=\frac{-5-3}{4}& m=\frac{1-3}{4}& m=\frac{-1-3}{4}\\ \Rightarrow & m=\frac{1}{2}& m=-2& m=-\frac{1}{2}& m=-1\end{array}$

Thus, the integral values of $m$ are $-1$ and $-2$.
Therefore, the number of integral values of $m$ are $2$

Hence, option(A) i.e. $2$ is correct.