Question

The graphs of 2x + 3y – 6 = 0, 4x – 3y – 6 = 0, x = 2 and y = $\dfrac{2}{3}$ intersect in
(a) 6 points
(b) 1 point
(c) 2 points
(d) No points
(e) An unlimited number of points

Answer 1

Hint: We are going to use the method of addition and then substitution for solving these equations. After this we will collect the points in order like (x, y) only. In case we are getting only one point then there is one intersecting point but, if we are getting more than one point after solving the equations then we will tick the option accordingly.

Complete step-by-step answer:
We will consider the equations 2x + 3y – 6 = 0...(i) and 4x – 3y – 6 = 0...(ii). Now we will add these two equations together. We will add that both the sides are added to each other like so, 2x + 3y – 6 + 4x – 3y – 6 = 0. By cancelling the common term – 3 from the left-hand side of the equation we will get 2x – 6 + 4x – 6 = 0. By further solving we will get 6x – 12 =0.
Now we will take the constant to the right side of the equation. Thus, we will get 6x = 12. As we know that the 6 divides 12 by 2 so, we will write x = 2. Now we will substitute this value in the equation (i). This results into 2x + 3y – 6 = 0 or, 2(2) + 3y – 6 = 0. Therefore, we will get 4 + 3y – 6 = 0.
After solving this equation, we will have 3y – 2 = 0 or, y = $\dfrac{2}{3}$. As we are only getting one point which is given by $\left( 2,\dfrac{2}{3} \right)$. The figure of the given question is shown below.

Hence, the correct option is (b).

Note: We can also directly get the answer by substituting the points one by one in the given equations and after this by looking at the completed graph we can have the answer. For example, if we consider the equation 2x + 3y – 6 = 0 we will put x = 0, 1, 2, 3. So, if we put x = 0 we will get 2(0) + 3y – 6 = 0 or, 3y – 6 = 0. Thus, we will get that y = 2. So, the point is (0, 2). Similarly, we will put the values in the equation and in other equations as well and trace the graph by the points that we will get. With the help of doing this we will be able to find the intersecting points clearly.