# Equivalence of systems of equations with solution zero.

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 $x_i=0? (1\le i\le n)$.Example set :

$x_1-x_2=0 ;$

$2x_1+x_2=0$

and

$3x_1+x_2=0 ;$

$x_1+x_2=0$

I got only zero as solution for both $x_1$ and $x_2$. Are these systems of equations equivalent?

• what does equivalence mean here? – piepi Jun 12 '16 at 16:50
• @piepi Look carefully: it's written in the first line. If and only if the equations of one are linear combinations of the equations of the other one and viceversa. – user228113 Jun 12 '16 at 16:51
• math.stackexchange.com/questions/46050/… – piepi Jun 12 '16 at 17:11

## 1 Answer

As Piepi points out, it depends on your definition of equivalence. Are two systems of equations defined to be "equivalent" 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)?

• 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. – Tanner Strunk Jun 12 '16 at 17:40
• I mean they are equivalent if they have the same solution. – Vikranth Inti Jun 13 '16 at 1:18
• Then haven't you answered your own question? If you define two equations as being equivalent if their solution sets are equivalent, and your two sets have equivalent solution sets of {0}, then by definition they are equivalent. – Tanner Strunk Jun 14 '16 at 21:04