Question

If the number of solutions of $3x-y=2$ and $9x-3y=6$ equations are $m$, then find $\frac{1}{m}$.

Answer 1

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 $\frac{a}{m}=\frac{b}{n}=\ne \frac{c}{d}$

Let's write given equation in the required form

$3x-y-2=0$ and $9x-3y-6=0$

Here, $a=3,b=-1,c=-2,m=9,n=-3,d=-6$

So in this case, Condition-2 is true i.e.

$\frac{3}{9}=\frac{-1}{-3}=\frac{-2}{-6}$ ...... $(=\frac{1}{3})$

Hence it has an infinite number of solutions.

So $m$=infinity, Hence $\frac{1}{m}=0$.