[Math] If two linear systems have the same solutions then the corresponding matrices are row equivalent

linear algebramatrices

Is the following statement true? 

If two linear systems have the same solutions then 
the corresponding matrices are row equivalent

I attempted a simple example with the systems 

5x + 2y = 23 
3x + 8y = 41

and 

2x + 3y = 18 
8x + 7y = 52

I was able to row reduce both of their augmented matrices into reduced row echelon form
\begin{bmatrix}1&0\\0&1\end{bmatrix}
Which I believe indicates row equivalence, however I don't know how I might prove the statement is true for all two same solution equivalent systems.
From my research I believe the following Theorem is critical for my understanding of the question enter image description here

Best Answer

HINT.-Draw four straight lines concurring at a point $P = (x, y)$. This point is a solution of the system formed by two of these four lines. But also it is solution of the system formed by the other two lines which can be arbitrary respect of the two first ones. What can you deduce of this?

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