Question

Find the number of integer values of m for which the x coordinate of the point of intersection of the line \[3x+4y=9\] and \[y=mx+1\] is an integer.

Answer 1

Hint: Using the equation \[y=mx+1\] , put the value of y in the equation \[3x+4y=9\] . Now, solve and get the value of x in terms of m. It is given that the value of m and x should be integers. Since x is an integer so, for integral values of x the possible values of the equation \[\left( 3+4m \right)\] can be any of 5, -5, 1 or 1. Now, solve one by one and get the values of m. Ignore those values of m which are not integers.

Complete step-by-step solution -

According to the question, we have the equation of the two lines intersecting at a point. We have to find the number of integral values of m for which the x coordinate of the point of intersection of these two lines is also an integer.
\[3x+4y=9\] ………………….(1)
\[y=mx+1\] ……………………(2)
First of all, we have to find the coordinates of the point of intersection of these two lines. To get the x coordinate of the point of intersection, we have to get the value of y from equation (2) and put it in equation (1).
Now, putting the value of y from equation (2) in equation (1), we get
\[3x+4\left( mx+1 \right)=9\]
On solving the above equation, we get
\[3x+4\left( mx+1 \right)=9\]
\[\begin{align}
  & \Rightarrow 3x+4\left( mx+1 \right)=9 \\
 & \Rightarrow 3x+4mx+4=9 \\
 & \Rightarrow \left( 3+4m \right)x=9-4 \\
 & \Rightarrow \left( 3+4m \right)x=5 \\
\end{align}\]
\[\Rightarrow x=\dfrac{5}{\left( 3+4m \right)}\] …………………(3)
In the question, it is given that the x coordinate of the point of intersection is an integer.
In equation (3), we have the value of x. Since it is given that the x coordinate of the point of intersection is an integer.
\[x=\dfrac{5}{\left( 3+4m \right)}\]
Since x is an integer so, for integral values of x the possible values of the equation \[\left( 3+4m \right)\] can be any of 5, -5, 1 or 1.
In case \[{{1}^{st}}\] , let us take \[\left( 3+4m \right)\] equal to 5.
\[\begin{align}
  & \left( 3+4m \right)=5 \\
 & \Rightarrow 4m=5-3 \\
 & \Rightarrow 4m=2 \\
 & \Rightarrow m=\dfrac{2}{4}=\dfrac{1}{2} \\
\end{align}\]
Here, we got \[m=\dfrac{1}{2}\] and \[\dfrac{1}{2}\] is not an integer.
Since it is given that the value of m should be an integer so \[m=\dfrac{1}{2}\] is not the solution of the equation \[\left( 3+4m \right)\] .
Now, in case \[{{2}^{nd}}\] , let us take \[\left( 3+4m \right)\] equal to -5.
\[\begin{align}
  & \left( 3+4m \right)=-5 \\
 & \Rightarrow 4m=-5-3 \\
 & \Rightarrow 4m=-8 \\
 & \Rightarrow m=\dfrac{-8}{4}=-2 \\
\end{align}\]
Here, we got \[m=-2\] and -2 is an integer.
Since it is given that the value of m should be an integer so \[m=-2\] is the solution of the equation \[\left( 3+4m \right)\] .
Now, in case \[{{3}^{rd}}\] , let us take \[\left( 3+4m \right)\] equal to 1.
\[\begin{align}
  & \left( 3+4m \right)=1 \\
 & \Rightarrow 4m=1-3 \\
 & \Rightarrow 4m=-2 \\
 & \Rightarrow m=\dfrac{-2}{4}=\dfrac{-1}{2} \\
\end{align}\]
Here, we got \[m=\dfrac{-1}{2}\] and \[\dfrac{-1}{2}\] is not an integer.
Since it is given that the value of m should be an integer so \[m=\dfrac{-1}{2}\] is not the solution of the equation \[\left( 3+4m \right)\] .
Now, in case \[{{4}^{th}}\] , let us take \[\left( 3+4m \right)\] equal to -1.
\[\begin{align}
  & \left( 3+4m \right)=-1 \\
 & \Rightarrow 4m=-1-3 \\
 & \Rightarrow 4m=-4 \\
 & \Rightarrow m=\dfrac{-4}{4}=-1 \\
\end{align}\]
Here, we got \[m=-1\] and -1 is an integer.
Since it is given that the value of m should be an integer so \[m=-1\] is the solution of the equation \[\left( 3+4m \right)\] .
Therefore, the integral values of are -2 and -1.
Hence, the number of integer values of m for which the x coordinate of the point of intersection of the line \[3x+4y=9\] and \[y=mx+1\] is an integer is 2.

Note: In this question, since \[m=\dfrac{1}{2}\] and \[m=\dfrac{-1}{2}\] is satisfying the equation \[\left( 3+4m \right)\] .Therefore, one might include \[m=\dfrac{1}{2}\] and \[m=\dfrac{-1}{2}\] as the solution of the equation \[\left( 3+4m \right)\] . This is contradiction because \[m=\dfrac{1}{2}\] and \[m=\dfrac{-1}{2}\] are not integers, and in the question it is given that m should be an integer.