Question

The number of integral values of m, for which the x-coordinate of the point of intersection of the line 3x+4y=9 and y=mx+1 is also an integer, is.
A.2
B.0
C.4
D.1

Answer 1

Hint: Solve both the equations to get the x-coordinate of the point of intersection in terms of m. Now using the relation you get, try to apply the constraints on m, including the ones given in the question and figure out the required values of m.

Complete step-by-step answer:
Let us start the question by finding the point of intersection of the lines given in the question. The equations of lines given in the question are:
3x+4y=9………………(i)
y=mx+1………………(ii)
So, if we substitute the value of y in terms of x, from equation (ii) in equation (i), we get
 3x+4y=9
\[~3x+4\left( mx+1 \right)=9\]
\[~\Rightarrow 3x+4mx+4=9\]
\[~\Rightarrow \left( 3+4m \right)x=5\]
\[~\Rightarrow x=\dfrac{5}{3+4m}\]
Now as it is given that x-coordinate must be an integer, so \[~\dfrac{5}{3+4m}\] must be an integer.
Therefore, the possible values of 3+4m are 1, -1, 5 and -5.
For 3+4m to be equal to 1, m must be equal to minus half, which is not an integer. Also, for 3+4m to be equal to 5, m must be 2, which is also not a possible case.
However, for 3+4m to be equal to -1 and -5 the values of m must be -1 and -2 respectively which are possible.
So, we can conclude that there are two integral values of m, for which the x-coordinate of the point of intersection of the line 3x+4y=9 and y=mx+1 is also an integer. Hence, the answer to the above question is option (A).

Note: Generally, in such questions where you are asked about the integral values, you generally have to reach the final answer through logic and statements as we did in the above question. There is no mathematical operation which will give you a direct answer. Also, try to keep the resultant expressions as clean and elaborative as possible, as they are the only ways of finding the constraints related to the asked values.