Is Consistency in Linear Systems Determined by Augmented Matrices?

[SylowTheorems]

Homework Statement

Quote from my textbook:

"The linear system Ax=b is consistent if and only if the number of nonzero rows of the augmented matrix [U| c], equals the number of nonzero rows in U."

[U,c] is the rref of [A,b]

Homework Equations

The Attempt at a Solution

I understand "only if" but not "if". "Only if" is true since the elementary row operations do not affect the solutions to the system and it is clear that a linear combination of zeros cannot equal a nonzero number. Can someone help me with "if"?

 

[rock.freak667]

number of nonzero rows in U is the rank of U,r(U)
the no. of nonzero rows of [U|c] is the rank of that matrix,r(U|c)

For a system of eq'ns to be constisent

$$r(U)=r(U|c) \leq n$$

n= no . of rows

 

[SylowTheorems]

Thanks for responding to my question!

Unfortunately, I am still confused. It seems like that just restated my question using the word "rank".

For a system of eq'ns to be constisent

$$r(U)=r(U|c) \leq n$$

I think the quote I gave came from a proof of this statement actually...

 

[rock.freak667]

well saying "if and only if", like restricting the statement so that the statement will be true when that condition is satisfied.

 

[SylowTheorems]

I figured it out. Thanks for your help.