Question

Considers the system of equations
$ax+by+cz=2$
$bx+cy+az=2$
$cx+ay+bz=2$
Where $a,b,c$ are real numbers such that $a+b+c=0$
Then the system

• A. Has two solutions
• B. Is inconsistent
• C. Has unique solutions
• D. Has infinitely many solutions

Answer 1

The correct option is B Is inconsistent

$\Delta =\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

$\Delta =a(bc-a^2)-b(b^2-ac)+c(ab-c^2)$

$\Delta =3abc-(a^3+b^3+c^3)$

$\Delta =3abc-\left[\right(a+b+c{)}^3-3ab(a+b)-3bc(b+c)-3ac(a+c)-6abc]$

$\Delta =3abc-[0-3ab(-c)-3bc(-a)-3ac(-b)-6abc]$ $(\because a+b+c=0)$

$\Delta =3abc-3abc=0$

No solution $\Rightarrow$ inconsistent