I understood that two systems of linear equations are equivalent if one can be obtained by the linear combination of the other system and vice versa. But can those two systems of equations be equivalent even if the solution x i = 0 ? ( 1 ≤ i ≤ n ).Example set: x 1 − x 2 = 0 ; 2 x 1 + x 2 = 0and 3 x 1 + x 2 = 0 ; x 1 + x 2 = 0Are these systems of equations equivalent?

Question

I understood that two systems of linear equations are equivalent if one can be obtained by the linear combination of the other system and vice versa. But can those two systems of equations be equivalent even if the solution       x    i    =  0  ?  (  1  ≤  i  ≤  n  ).Example set:      x    1    −      x    2    =  0  ;  2      x    1    +      x    2    =  0and  3      x    1    +      x    2    =  0  ;      x    1    +      x    2    =  0Are these systems of equations equivalent?

Janiyah Patton · Accepted Answer

It depends on your definition of equivalence. Are two systems of equations defined to be &quot;equivalent&quot; if their solution sets are equivalent as sets? Or are they ONLY equivalent if one system can be obtained by a linear combination of the other system (and vice versa)?

Brock Byrd · Accepted Answer

Note that the latter equivalence of systems by linear combinations naturally implies the former equivalence of their solution sets, but having equivalent solution sets does NOT imply that one system can be obtained by a linear combination of the other system.