Ax=b1 and Ax=b2 are consistent. Is the system Ax=b1+b2 necessarily consistent?

[iamzzz]

Homework Statement

Suppose A is m*n matrix b1 and b2 are m*1 vector and the systems Ax=b1 and Ax=b2 are consistent. Is the system Ax=b1+b2 necessarily consistent?

Homework Equations

The Attempt at a Solution

I think Ax = b1 + b2 should be consistent but i don't know how to prove..

 

[Dick]

Homework Statement

Suppose A is m*n matrix b1 and b2 are m*1 vector and the systems Ax=b1 and Ax=b2 are consistent. Is the system Ax=b1+b2 necessarily consistent?

Homework Equations

The Attempt at a Solution

I think Ax = b1 + b2 should be consistent but i don't know how to prove..

No thoughts about how to prove at all? A(x1+x2)=Ax1+Ax2. Think about it some more.

 

[iamzzz]

No thoughts about how to prove at all? A(x1+x2)=Ax1+Ax2. Think about it some more.

let x1 be the solution of Ax1=b1 and x2 be the solution of Ax2=b2

So Ax1+Ax2=b1+b2=A(x1+x2)
so (x1+x2)could be x
prove ?
Is this correct ?

 

[Dick]

let x1 be the solution of Ax1=b1 and x2 be the solution of Ax2=b2

So Ax1+Ax2=b1+b2=A(x1+x2)
so (x1+x2)could be x
prove ?
Is this correct ?

I would say let x1 be ANY solution to Ax=b. There may be more than one. But why are you asking "Is this correct?". What part of it are you worried about?

 

[iamzzz]

I would say let x1 be ANY solution to Ax=b. There may be more than one. But why are you asking "Is this correct?". What part of it are you worried about?

I mean does that prove the problem ?

Anyway thanks for the help

 

[Dick]

I mean does that prove the problem ?

Anyway thanks for the help

I'm just saying I would feel better if you KNEW it solved the problem instead of having to ask. Yes, it's fine.