Question

Assertion

The linear equations $x - 2y - 3 = 0$ and $3x + 4y - 20 = 0$ have exactly one solution.

Reason

The linear equations $2x + 3y - 9 = 0$ and $4x + 6y - 18 = 0$have a unique solution

A. Both assertion and reason are correct and the reason is the correct explanation for assertion

B. Both assertion and reason are correct but the reason is not the correct explanation for assertion

C. The assertion is correct but the reason is incorrect

D. The assertion is incorrect but the reason is correct

Answer 1

We have been given two different statements in the assertion and the reason so we’ll check for them separately, in both the cases we’ll find the number of solutions for the given linear equations. After that, we’ll check with the given options.

Complete step by step answer:

for assertion

solving for the linear equations $x - 2y - 3 = 0$ and $3x + 4y - 20 = 0$
$\Rightarrow 3x + 4y - 20 = 0.............(i)$
$\Rightarrow x - 2y - 3 = 0...............(ii)$
$\Rightarrow x = 2y + 3$
Substituting the value of ‘x’ in equation(i)
$\Rightarrow 3(2y + 3) + 4y - 20 = 0$
Simplifying the brackets
$\Rightarrow 6y + 9 + 4y - 20 = 0$
Simplifying the like terms
$\Rightarrow 10y - 11 = 0$
$\Rightarrow y = \dfrac{{11}}{{10}}$
Substituting the value of ‘y’ in equation(ii)
$\Rightarrow x - 2\left( {\dfrac{{11}}{{10}}} \right) - 3 = 0$
Simplifying by grouping the like terms
$\Rightarrow x = 2\left( {\dfrac{{11}}{{10}}} \right) + 3$
$\therefore x = \dfrac{{26}}{5}$
Since we got a single solution i.e. $\left( {\dfrac{{11}}{{10}},\dfrac{{26}}{5}} \right)$
Therefore the given equations have only one solution

for reason

solving for the linear equations $2x + 3y - 9 = 0$ and $4x + 6y - 18 = 0$
$\Rightarrow 2x + 3y - 9 = 0..........(iii)$
$\Rightarrow 4x + 6y - 18 = 0...........(iv)$
Dividing equation(iv) by 2
$\Rightarrow 2x + 3y - 9 = 0$
From the above equations, we can say that both the equations of the line are the same or we can say that both the line coinciding
Since the lines are coinciding, then we’ll have an infinite number of solutions.
Now, from the above results, we can say that assertion is correct but reason id not true
Option(C) is correct.

Note: We can also verify our solution with the help of graph plotting

Assertion

Reason
Since both lines coincide,