# Determine (without doing row operations) that a system is consistent

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Hello,

I just started learning Elementary Linear Algebra. I've read through the first chapter in my book (Elementary Linear Algebra by Larson, Edwards, Falvo, 5th edition). There were two "Discovery" questions that try to see if you know how systems of equations work. I am unsure on both of them.

These are not homework questions, rather a test to see if you know how to determine if a system is consistent or if a system has an infinite # of solutions.

Consider the system of linear equations.
2x1 + 3x2 + 5x3 = 0
-5x1 + 6x2 -17x3 = 0
7x1 - 4x2 + 3x3 = 0

Without doing any row operations, explain why this system is consistent.

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The following system has more variables than equations. Why does it have an infinite number of solutions?
2x1 + 3x2 + 5x3 + 2x4 = 0
-5x1 + 6x2 -17x3 - 3x4 = 0
7x1 - 4x2 + 3x3 + 13x4 = 0

Since I am just beginning to learn linear algebra, I am not sure exactly the answer for either question. I know a system is consistent if it has exactly one solution or if it has infinite solutions, so how could I tell if the system for question one is consistent without doing any row operations?

For the second problem, I know that a system has infinite solutions if after doing row operations you get a free variable, but how do you determine if a variable is considered free?