I was recently solving a system of linear equations, 3 equations and 3 unknowns. I first solved via Row Reduction of the matrix and got a valid answer, but my friend attempted to solve the system using informal algebra methods and got the wrong answer. I know his answer is wrong, but I am struggling to explain what mathematical rule he broke.
Here is the system:
Combining the first and third equation, one gets $x=-2y$. Plugging this back into equation one, one gets $z=-y$. Setting $x=1$, one gets the vector $<1,-1/2,1/2>$. This vector is valid for equations 1 and 3, but not for equation 2.
Now I know that this is not the proper technique for solving a system of three variables and that equation 2 was not used so how should one expect it to be satisfied. I know that this solution is wrong, but I am unsure how to explain what is wrong about it other than saying "that's not the way it's done." I personally made this mistake when first learning linear algebra and "that's not the way it's done" is all my teacher could say. If anyone has a better explanation for what exactly is wrong about this, I would greatly appreciate. Also, since equation 2 is not being used, it is 2 equations, 3 unknowns so there should be 2 free variables, not one (again I think this will occur if elimination is "done correctly").