iamzzz said: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 said:No thoughts about how to prove at all? A(x1+x2)=Ax1+Ax2. Think about it some more.
iamzzz said: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 mean does that prove the problem ?Dick said: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 said:I mean does that prove the problem ?
Anyway thanks for the help
Yes, it is possible for both equations to be consistent. This means that there is at least one solution that satisfies both equations simultaneously.
A system of equations is consistent if and only if it has at least one solution. This can be determined by solving the equations and checking if there is a set of values that satisfies all the equations.
A system of equations is inconsistent if there is no set of values that satisfies all the equations. In other words, the equations are contradictory and cannot be solved simultaneously.
Not necessarily. Just because both equations are consistent does not mean that their sum will also be consistent. It is possible for b1 and b2 to have different values that do not result in a consistent solution when added together.
No, a consistent system of equations must have at least one solution. If there is no solution, then the system is inconsistent.