Question

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

• A. $2$
• B. $0$
• C. $4$
• D. $1$

Answer 1

The correct option is A

$2$

Explanation for the correct option:

Step 1: Find the possible values of $x$.

Two equations $3x+4y=9$ and $y=mx+1$.

To find the $x-$coordinate of the point of intersection of the given two lines.

From both the given equation, we get.

$$\begin{gathered} 3x+4(mx+1)=9 \\ \Rightarrow 3x+4mx+4=9 \\ \Rightarrow 3x+4mx=5 \\ \Rightarrow \left(3+4m\right)x=5 \\ \Rightarrow x=\frac{5}{\left(3+4m\right)}...1 \end{gathered}$$

Since, $x$ is an integer. So, the possible values of $x$ are $-5,-1,1$ and $5$.

Step 2: Find the integer values of $m$.

When $x=-5$.

$$\begin{gathered} -5=\frac{5}{\left(3+4m\right)} \\ \Rightarrow 3+4m=-1 \\ \Rightarrow 4m=-4 \\ \Rightarrow m=-1 \end{gathered}$$

Therefore, the values of $m$ when $x=-5$ is $-1$.

When $x=-1$.

$$\begin{gathered} -1=\frac{5}{\left(3+4m\right)} \\ \Rightarrow 3+4m=-5 \\ \Rightarrow 4m=-8 \\ \Rightarrow m=-2 \end{gathered}$$

Therefore, the values of $m$ when $x=-1$ is $-2$.

When $x=1$.

$$\begin{gathered} 1=\frac{5}{\left(3+4m\right)} \\ \Rightarrow 3+4m=5 \\ \Rightarrow 4m=2 \\ \Rightarrow m=\frac{1}{2} \end{gathered}$$

Therefore, the values of $m$ when $x=1$ is $\frac{1}{2}$.

When $x=5$.

$$\begin{gathered} 5=\frac{5}{\left(3+4m\right)} \\ \Rightarrow 3+4m=1 \\ \Rightarrow 4m=-2 \\ \Rightarrow m=-\frac{1}{2} \end{gathered}$$

Therefore, the values of $m$ when $x=5$ is $-\frac{1}{2}$.

Thus, the number integer values of $m$ when the $x-$coordinate of the point of intersection of the given lines is also an integer is $2$.

Hence, option $A$ is the correct answer.