How do I turn this question into equations and then solve the system? Bachelle wants to know the density of her bracelet, which is a mix of gold...

The following method does not use a system of two or three
equations but rather a work/mixture approach:

Since the
volume of gold is twice the volume of the silver, we can use the ratio of 2G/1S to
calculate the overall density. 

The
d of G is 19.3g/cc and the d
of S is 10.5g/cc.  So now I take 2G=2*19.3=38.6 and then add 10.5 to get a total of
49.1g/cc. 

But this is 3 times the total density, because
we added the densities of two golds and one silver.  So, dividing 49.1 by 3, we get
16.36666... or, rounding to two decimal places, 16.37
g/cc.

Make sure not to confuse mass and
weight.  Weight takes into account the force of gravity; mass does
not.