Question

The graphs of $2x + 3y - 6 = 0,4x - 3y - 6 = 0{\text{ ,}}x = 2{\text{ and }}y = \dfrac{2}{3}$ intersect in
A.Four points
B.One point
C.In no points
D.Infinite number of points

Answer 1

Hint: In order to solve this question we need to find the point of intersection of these lines. We will find the point of intersection of the first two lines and then the third line with others.

Complete step-by-step answer:
Given
\[
  2x + 3y - 6 = 0......................(1) \\
  4x - 3y - 6 = 0......................(2) \\
  x = 2......................................(3) \\
  y = \dfrac{2}{3}....................................(4) \\
 \]
Adding and solving equation (1) and (2) we get
\[
   \Rightarrow2x + 3y - 6 + 4x - 3y - 6 = 0 \\
   \Rightarrow 6x = 12 \\
   \Rightarrow x = 2 \\
 \]
Substituting the value of x in equation (1), we get
\[
   \Rightarrow 2 \times 2 + 3y - 6 = 0 \\
   \Rightarrow 3y = 2 \\
   \Rightarrow y = \dfrac{2}{3} \\
 \]
Hence the point of intersection of these two lines is $\left( {2,\dfrac{2}{3}} \right)$
Also $x = 2$ and $ y = \dfrac{2}{3}$ passes through the same point of intersection $\left( {2,\dfrac{2}{3}} \right)$ .
Hence, all passes through only one point of intersection.
So, option B is the correct option.

Note: In order to solve these types of questions, remember the basic concept of solving the equations such as elimination method, cross multiplication method and substitution method. Also remember the concept of slope and equations of straight lines. There are five equations of straight lines such as slope intercept form. These types of problems can also be solved by the method of graphs, but it is rather more simpler by the method of algebra.