Question 1204100
 Consider the system of equations
{{{ x + 2y - 3z = a}}} 
{{{ 2x + 6y - 11z = b}}} 
{{{ x - 2y + 7z = c}}} 
where {{{ a}}} , {{{ b }}} and {{{ c }}} are three real numbers.



Solution:

{{{ x + 2y - 3z = a}}} 
{{{ 2x + 6y - 11z = b}}} 
{{{ x - 2y + 7z = c}}} 


Δ={{{ matrix(3,3,
1,2,-3,
2,6,-11,
1,-2,7)}}} ={{{0}}} 


 Δ1={{{ matrix(3,3,
a, 2, -3,
b, 6,-11,
c, -2,7)}}} ={{{4(5a-2b-c)}}} 


Δ2={{{ matrix(3,3,
1, a,-3,
2,b,-11,
1,c,7)}}} ={{{-5(5a-2b-c) }}} 


Δ3={{{ matrix(3,3,
1,2,a,
2,6,b,
1,-2,c)}}} ={{{-2(5a-2b-c) }}} 


 If {{{ 5a=2b+c }}} => Δ1=Δ2=Δ3={{{0}}} => system will have infinitely many solutions .