Question

Consider the following statements. The system of equations
$2x-y=4$
$px-y=q$
1. has a unique solution if $p\ne 2$
2. has infinitely many solutions if $p=2,q=4$
Of these statement

• A. 1 alone is correct
• B. 2 alone is correct
• C. 1 and 2 are correct
• D. 1 and 2 are false

Answer 1

The correct option is C 1 and 2 are correct

Given equations are $2x-y=4---eqn\left(1\right)$ and $px-y=q---eqn\left(2\right)$

If $p=2,$So $eqn\left(2\right)$ becomes $2x-y=q$

Using conditions of consistency for linear equations $ax+by+c=0$ and $mx+ny+d=0$ we have :

$1.$ System of linear equations is consistent with unique solutions if $\frac{a}{m}\ne \frac{b}{n}$

$2.$ System of linear equations is consistent with infinitely many solutions if $\frac{a}{m}=\frac{b}{n}=\frac{c}{d}$

$3.$ System of linear equations is inconsistent i.e it has no solution if $\frac{a}{m}=\frac{b}{n}\ne \frac{c}{d}$

So we get that when $p=2$ then $\left(1.\right)$ is satisfied .Similarly when p=2 and $q=4$ the $\left(2.\right)$ is satisfied so it will have infinitely many solutions