Question

The system of equations
$x+y+z=4$
$2x+5y-2z=3$
$x+7y-7z=5$ is

• A. consistent
• B. trivial
• C. infinite solutions
• D. inconsistent

Answer 1

The correct option is D inconsistent

For given system of equations Augumented matrix is $\left[\begin{array}{cccc}1& 1& 1& 4\\ 2& 5& -2& 3\\ 1& 7& -7& 5\end{array}\right]$

$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 2& 5& -2\\ 1& 7& -7\end{array}\right|=1(-35+14)-1(-14+2)+1(14-5)=-21+12+9=0$

${\Delta }_1=\left|\begin{array}{ccc}4& 1& 1\\ 3& 5& -2\\ 5& 7& -7\end{array}\right|=4(-21)-1(-11)+1(-4)=-77\ne 0$

As, one of ${\Delta }_1,{\Delta }_2,{\Delta }_3$ is non-zero and $\Delta =0$

given system has no solution.

Hence, option D.