Can Ax=0 Be Inconsistent? Solving Linear Equations with a Zero Constant

[Tryingtolearn]

Hopefully someone can help me out with this one. I was in a conversation and the topic came up on what can an can not be inconsistent.

Assume that A, x, and 0 are all of the corrent order for Ax=0. According to everything I have learned, Ax=b if b=0 then it is always consistent. Does anyone know when your are solving a system of linear equations if there is every a case when b=0 and sill be inconsistent?

Your help is much appreciated.

 

[arildno]

If x=0, what is then Ax?

 

[M98Ranger]

In order for Ax=0 to be inconsistent, there must be no solution or an infinite number of solutions for the system of equations. This means that there is no unique solution that satisfies all the equations in the system. In other words, there is no value of x that can make all the equations true simultaneously.

If b=0 in the system Ax=b, then there is always a solution. This is because when b=0, the system becomes Ax=0 which is a special case where there is always a unique solution, which is x=0. This is because any number multiplied by 0 is always equal to 0. Therefore, Ax=0 can never be inconsistent when b=0.

However, if b≠0, then it is possible for the system to be inconsistent. This is because when b≠0, the system becomes Ax=b and there may not be a value of x that can satisfy all the equations. In this case, the system is inconsistent and has no solution.

In summary, Ax=0 can only be inconsistent when b≠0 in the system Ax=b. When b=0, there is always a unique solution and the system is consistent. So, to answer the question, yes, there are cases when b=0 and the system is still inconsistent, but this only happens if there are other variables in the system besides x.