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

by iamzzz
Tags: axb1, axb2, consistent, necessarily
 P: 22 1. The problem statement, all variables and given/known data 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? 2. Relevant equations 3. The attempt at a solution I think Ax = b1 + b2 should be consistent but i dont know how to prove..
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P: 25,228
 Quote by iamzzz 1. The problem statement, all variables and given/known data 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? 2. Relevant equations 3. The attempt at a solution I think Ax = b1 + b2 should be consistent but i dont know how to prove..
No thoughts about how to prove at all? A(x1+x2)=Ax1+Ax2. Think about it some more.
P: 22
 Quote by Dick 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 ?

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P: 25,228
Ax=b1 and Ax=b2 are consistent. Is the system Ax=b1+b2 necessarily consistent?

 Quote by iamzzz 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?
P: 22
 Quote by Dick 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