How to solve equation that intersects line with surface of normal density function

Hi guys,

I'm trying to figure out how to solve intersection between a line and the surface of a normal density function at a certain probability (which I assume forms ellipsoid). I've reached this stage where I have the equation for the surface of the density

And I have the equation the line in the format

I need to apply this an arbitray number of dimensions, so I can't solve the equations by hand and then plug the values :/

Do you know how to intersect this sequence equations in matlab? I looked at solve but I couldn't figure out how to convert the first equation in the required format... :/

My background in Math and Matlab is kinda limited, so any help would be great!

Thanks!