Question

Tick the correct option.
$\dfrac{x}{y} = \dfrac{2}{3}$
(A). Infinitely many solutions
(B). Unique solution
(C). No solution
(D). Only two solutions

Answer 1

- Hint- in order to solve the problem either try for different values or rather convert it into some coordinate geometry problem of some figure.

Complete step-by-step solution -

Given equation is $\dfrac{x}{y} = \dfrac{2}{3}$
Let us simplify the equation by cross multiplying and try to separate the variables.
\[
   \Rightarrow \dfrac{x}{y} = \dfrac{2}{3} \\
   \Rightarrow 3x = 2y \\
   \Rightarrow y = \dfrac{3}{2}x \\
 \]
As we get the equation as $y = \dfrac{3}{2}x$
Also we know the general equation of the line is $y = mx + c$
So the following equation represents a line where
$m = \dfrac{3}{2}\& c = 0$
Also we know that a line has infinitely many points or infinitely many solutions so the above equation has infinitely many solutions.
Some of the solutions can be found out by putting some value of x and finding the corresponding value of y.
$
  x = 0,y = 0 \\
  x = 4,y = 6 \\
  x = 6,y = 9 \\
 $
And many more.
Hence, the above equation has infinitely many solutions.
So, option A is the correct option.

Note- In order to solve such a type of problem. The method of the coordinate geometry is one of the easiest. These types of problems can also be solved by direct substitution and checking of results by putting some random values.