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[SOLVED] how you know this system of linear equations has infinitely many solutions

karush

Well-known member
Jan 31, 2012
3,066
The details for Archetype J (System with
six equations,
nine variables.
Consistent.
Null space of coefficient matrix has dimension 5.)

include several sample solutions.

Verify that one of these solutions is correct (any one, but just one).
Based only on this evidence, and especially without doing any row operations,
explain how you know this system of linear equations has infinitely many solutions

Ok the only thing I can think of the there is more variables than equations so you cannot have a unique solution
also I didn't know exactly what "Null space of coefficient matrix has dimension 5" meant
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,273
1) Say you have two lines \(\displaystyle y = ax + b\) and \(\displaystyle y = 3x - 7\). How many solutions does this have?

2) Say you have two lines \(\displaystyle y = ax + b\) and \(\displaystyle y = -ax - b\). How many solutions does this have?

3) Say you have two lines \(\displaystyle y = ax + b\) and \(\displaystyle y = ax\). (\(\displaystyle b \neq 0\).) How many solutions does this have?

Now take a look at the coefficient matrix for each. What is the determinant of each? What does that tell you?

-Dan