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How can you prove that the normal equations: $(X^TX)\beta = X^TY$ have one or more solutions without the assumption that X is invertableinvertible?

My only guess is that it has something to do with generalized inverse, but I am totally lost.

How can you prove that the normal equations: $(X^TX)\beta = X^TY$ have one or more solutions without the assumption that X is invertable?

My only guess is that it has something to do with generalized inverse, but I am totally lost.

How can you prove that the normal equations: $(X^TX)\beta = X^TY$ have one or more solutions without the assumption that X is invertible?

My only guess is that it has something to do with generalized inverse, but I am totally lost.

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Question about a normal equation proof

How can you prove that the normal equations: $(X^TX)\beta = X^TY$ have one or more solutions without the assumption that X is invertable?

My only guess is that it has something to do with generalized inverse, but I am totally lost.