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 (1) (2)(3) (4)

• A. $0$
• B. $2$
• C. $4$
• D. $1$

Answer 1

The correct option is B

$2$

Solve the given equations for $x$

Given equations are

$$\begin{gathered} 3x+4y=9\_\_\_\_\_\_\_\_\_\_\left(1\right) \\ y=mx+1\_\_\_\_\_\_\_\_\_\_\_\left(2\right) \end{gathered}$$

Substitute the equation $\left(2\right)$ in equation$\left(1\right),$ we get

$$\begin{gathered} 3x+4(mx+1)=9 \\ 3x+4mx+4=9 \\ 3x+4mx=5 \\ x(3+4m)=5 \\ x=\frac{5}{(3+4m)} \\ LHS=x \\ RHS=\frac{5}{(3+4m)} \end{gathered}$$

$x$ has to be an integer. So the possible values of$x$ are $5,-5,1,-1$.

$\therefore$ The denominator on R.H.S. has to be $1,-1,5,-5$ respectively.

So $3+4m=1$

$m=-\frac{1}{2}$(not integral)

$$\begin{gathered} 3+4m=-1 \\ \Rightarrow m=-1 \end{gathered}$$

$3+4m=5$

$m=\frac{1}{2}$(not integral)

$$\begin{gathered} 3+4m=-5 \\ \Rightarrow m=-2 \end{gathered}$$

We get $2$ integral values for$m.$

Hence, two integral values of$m$ are possible.