Question

If the equations
$(b+c)x+(c+a)y+(a+b)=0,cx+ay+b=0$ and $ax+by+c=0$ are consistent, then show that either $a+b+c=0$ or $a=b=c$.

Answer 1

For consistency of three equations in two unknown. we must have D = $0$
$△=\left|\begin{array}{ccccc}b+c& c+a& a+b//c& a& b\\ a& b& c& & \end{array}\right|=0$
or
$\left|\begin{array}{ccc}b& c& a\\ c& a& b\\ a& b& c\end{array}\right|=0$
by $R_1-R_2$
or $3abc-a^3-b^3-c^3=0$
or $a^3+b^3+c^3-3abc=0$
Hence either $a+b+c=0$ or $b-c=0,c-a=0,a-b=0,i.e.,a=b=c.$ Hence the conditions are either $a+b+c=0ora=b=c$