Question

Consider the following system of equations:
$x+2y-3z=a$
$2x+6y-11z=b$
$x-2y+7z=c$,
Where $a,b$ and $c$ are real constants. Then the system of equations:

• A. has a unique solution when
  $5a=2b+c$
• B. has infinite number of solutions when $5a=2b+c$
• C. has no solution for all $a,b$ and $c$
• D. has a unique solution for all $a,b$ and $c$

Answer 1

The correct option is B has infinite number of solutions when $5a=2b+c$

$\Delta =\left|\begin{array}{ccc}1& 2& -3\\ 2& 6& -11\\ 1& -2& 7\end{array}\right|$
$=20-2\left(25\right)-3(-10)$
$=20-50+30=0$

${\Delta }_1=\left|\begin{array}{ccc}a& 2& -3\\ b& 6& -11\\ c& -2& 7\end{array}\right|$
$=20a-2(7b+11c)-3(-2b-6c)$
$=20a-14b-22c+6b+18c$
$=20a-8b-4c$
$=4(5a-2b-c)$

${\Delta }_2=\left|\begin{array}{ccc}1& a& -3\\ 2& b& -11\\ 1& c& 7\end{array}\right|$
$=7b+11c-a\left(25\right)-3(2c-b)$
$=7b+11c-25a-6c+3b$
$=-25a+10b+5c$
$=-5(5a-2b-c)$

${\Delta }_3=\left|\begin{array}{ccc}1& 2& a\\ 2& 6& b\\ 1& -2& c\end{array}\right|$
$=6c+2b-2(2c-b)-10a$
$=-10a+4b+2c$
$=-2(5a-2b-c)$

for infinite solution
$\Delta ={\Delta }_1={\Delta }_2={\Delta }_3=0$
$\Rightarrow 5a=2b+c$