Finding equation for multiple regression slopes in Excel's LINEST() function

I have a client who has an issue getting the LINEST() function to work for them.  This is my first time using LINEST(), and what seemed, at first, to be straightforward has deepened considerably in levels of complexity.  I am an accomplished VBA programmer, and I feel the best way to give the client what they need is to duplicate the equation in VBA to calculate the slope (m) when there are multiple X Axes.  When only one X axis exists, I have validated LINEST against my own independent calculation, as shown by the equation below.


I am looking for a similar equation I can apply, given 2 or more X Axes.  Reproducing the underlying query in VBA will make for a much more versatile function, capable of calculating LINEST in a more wide-ranging set of circumstances than Excel currently allows, thus meeting my client's requirements.  For those not familiar with LINEST(), I have included a couple of sites that have given me the best start on solving the problem.  The office support site provided the equation, as shown above, which gave me a reliable means to have an independent check against the Excel function on a single X axis.  Every site I've seen thus far provides the LINEST() equation for multiple X Axes, but nowhere have I seen a more nuanced formula that tells me what Excel is doing to derive that number.  That's my missing piece.

Thanks, in advance, for your support.  ~ Peter Ferber

http://www.mit.edu/~mbarker/formula1/f1help/04-g-m60.htm
https://support.office.com/en-us/article/LINEST-function-84d7d0d9-6e50-4101-977a-fa7abf772b6d