Is the following statement correct? How can I prove it?

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> $Ax=b$ has a solution if and only if $A^TAx=A^Tb$ has a solution.
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> 
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I know that $Ax=b$ has a solution if and only if $A$ is invertible. In that case, the solution is $A^{-1}b$. Therefore, every solution of $Ax=b$, is also a solution of $A^TAx$. But this is not the other way around, right? Or is it?

Any help appreciated! Thank you!

(I know there are other similar questions about this, but these posts do not answer my specific question)