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 Reason is the correct explanation for Assertion.
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
C. Assertion is correct but Reason is incorrect.
D. Assertion is incorrect but Reason is correct.

Answer 1

Hint: A system of linear equations can have any of the three possible types of solutions: Unique Solution, Infinitely many Solutions and No Solution. If the lines are intersecting at one point, then the linear equations have a unique solution. If the lines are coincident, then the system has infinitely many solutions. And, if the lines are parallel, then the system does not have any solution.

Complete step-by-step answer:
Let us first consider the assertion. It says that the linear equations $x-2y-3=0$ and $3x+4y-20=0$ have exactly one solution.
Let $x-2y-3=0$ --- (1)
And, $3x+4y-20=0$ ---(2)
In order to solve these equations, let us multiply the first equation by 3.
$3(x-2y-3)=3 \times 0$
$\implies 3x-6y-9=0$ ----(3)
Subtracting equation 3 from equation 2, we get,
$3x+4y-20-(3x-6y-9)=0$
Removing the brackets, we get,
$3x+4y-20-3x+6y+9=0$
$\implies 10y-11=0$
$\implies y=\dfrac{11}{10}$
Now, in order to find the value of x, substituting the value of y in equation 1, we get,
$x-2\times \dfrac{11}{10}-3=0$
$\implies x=\dfrac{22}{10}+3=\dfrac{22+30}{10}=\dfrac{52}{10}=\dfrac{26}{5}$
Thus, the pair of linear equations given possess exactly one solution (unique solution).
Hence, the assertion is correct.
Now, let us consider the reason. It says that the linear equations $2x+3y-9=0$ and $4x+6y-18=0$ have a unique solution.
Let $2x+3y-9=0$ ---(1)
And, $4x+6y-18=0$ ---(2)
In order to solve these equations, let us multiply the first equation by 2.
$2(2x+3y-9)=2 \times 0$
$\implies 4x+6y-18=0$ ---(3)
As, equation 2 and 3 are same thus, thus the two linear equations given to us are coincident possessing infinitely many solutions.
Thus, the reason is not correct.
Thus, Assertion is correct but the Reason is incorrect.
Hence, option C is correct.

Note: The question given to us can be asked in other forms too. They may give us the linear equations and ask to find the unique solution, if it exists. In that case, we first need to check if the equations possess a unique solution. If they do, then by finding the point of intersection, one can get a solution to the equations.