I would like to number equations similarly to how they do in papers i.e. the equation centred and the number right justified.



i.e.

Is there any way to do this in Word or am I forced to use Latex if I'm going to make a professional looking paper?

There is possibly a way with tabs but these can be fiddly. However, you could use a table - one row, 2 columns with column 1 wide and centred and column 2 (on the right) narrow - and remove the gridlines afterwards so you can't see it's a table.
Use a 4 column table if you want the equations centred on the = sign, col 1 right justified, = in col 2, col 3 left justified col 4 for your numbers. Adjust column widths until they look ok and then remove the gridlines. Copy and paste the table for further equations so they all look the same. I have used this method for equations. Hope this helps.