Question

Solve the following sets of equations using Cramer's rule and remark about their consistency.

$x+y+z-6=0$
$2x+y-z-1=0$
$x+y-2z+3=0$

• A. $x=1,y=2,z=3$; consistent
• B. $x=2,y=1,z=3$; Consistent
• C. Inconsistent
• D. None of the above

Answer 1

The correct option is A $x=1,y=2,z=3$; consistent

Given,

$x+y+z-6=0$

$2x+y-z-1=0$

$x+y-2z+3=0$

$\left|\begin{array}{ccc}1& 1& 1\\ 2& 1& -1\\ 1& 1& -2\end{array}\right|$

$=1(-2+1)-1(-4+1)+1(2-1)$

$\Delta =3$

$\left|\begin{array}{ccc}6& 1& 1\\ 1& 1& -1\\ -3& 1& -2\end{array}\right|$

$=6(-2+1)-1(-2-3)+1(1+3)$

${\Delta }_1=3$

$\left|\begin{array}{ccc}1& 6& 1\\ 2& 1& -1\\ 1& -3& -2\end{array}\right|$

$=1(-2-3)-6(-4+1)+1(-6-1)$

${\Delta }_2=6$

$\left|\begin{array}{ccc}1& 1& 6\\ 2& 1& 1\\ 1& 1& -3\end{array}\right|$

$=1(-3-1)-1(-6-1)+6(2-1)$

${\Delta }_3=9$

$x=\frac{{\Delta }_1}{\Delta }=\frac{3}{3}=1$

$y=\frac{{\Delta }_2}{\Delta }=\frac{6}{3}=2$

$z=\frac{{\Delta }_3}{\Delta }=\frac{9}{3}=3$