Question

Assertion :If the system of equations $2x+3y=7$ and $2ax+(a+b)y=28$ has infinitely many solutions, then $2a-b=0$ Reason: The system of equations $3x-5y=9$ and $6x-10y=8$ has 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

The correct option is C Assertion is correct but Reason is incorrect

Assertion, given system of equations has infinitely many solutions if
$\frac{2}{2a}=\frac{3}{a+b}=\frac{-7}{-28}$

$\Rightarrow \frac{1}{a}=\frac{3}{a+b}=\frac{1}{4}\Rightarrow 3a=a+b\Rightarrow 2a-b=0\Rightarrow b=2a$

Also clearly $a=4$ and $a+b=12\Rightarrow b=8$

$\therefore 2a-b=8-8=0$ $\therefore$ Assertion is true
Reason

The pair of equations are of the form $a_{1}x+b_{1}y=c_1$ and $a_{2}x+b_{2}y=c_2$

Here $a_1=3,b_1=-5,c_1=9$ and $a_2=6,b_2=-10,c_2=8$

A pair of equations will have a unique solution, if $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

But reason is false $\because \frac{3}{6}=\frac{-5}{-10}$