Question

If $x=a,y=b,z=c$ is the solution of the system of simultaneous linear equations $x+y+z=4$, $x-y+z=2,x+2y+2z=1$, then $ab+bc+ca=$

• A. 0
• B. -25
• C. 1
• D. -4

Answer 1

Answer: (B)

Given that,
$x+y+z=4$, $x-y+z=2$ and $x+2y+2z=1$
$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1& -1& 1\\ 1& 2& 2\end{array}\right|=l(-2-2)-1(2-1)+(2+1)]$
$=-4-1+3=-2$
${\Delta }_1=\left|\begin{array}{ccc}4& 1& 1\\ 2& -1& 1\\ 1& 2& 2\end{array}\right|4(-2-2-1(4-1)+1(4+1)$
$=4(-4)-3+5=-16-3+5=-14$
${\Delta }_2=\left|\begin{array}{ccc}1& 4& 1\\ 1& 2& 1\\ 1& 1& 2\end{array}\right|$
${\Delta }_2=1(4-1)-4(2-1)+1(1-2)=3-4-1=-2$
${\Delta }_3=\left|\begin{array}{ccc}1& 1& 4\\ 1& -1& 2\\ 1& 2& 1\end{array}\right|$
${\Delta }_3=1(-1-4)-1(1-2)+4(2+1)=(-5)+1+12=8$
$x=\frac{{\Delta }_1}{\Delta }=\frac{-14}{-2}=7$,
$y=\frac{{\Delta }_2}{\Delta }=\frac{-2}{-2}=1$
and $z=\frac{{\Delta }_3}{\Delta }=\frac{8}{-2}=-4$
So, $a=7,b=1,c=-4$
Now, $ab+bc+ca$
$=7(1)+1(-4)+(-4)(7)=7-4-28=-25$