Linear Algebra Word Problem I have been this for hours, yes it is a homework question, but did not turn it in last weeks when I was suppose to, but I have a test coming up, so gotta figure this out not sure how to formulate questions using a system of equations
The density of gold is $19.3g/cm^3$ and the density of silver is $10.5g/cm^3$ a certain crown is made entirely from silver and gold if the total volume of the grown is $220cm^3$, and the weight of the crown is $3823.6g$, what percentage of the total mass is gold?
I am not sure how to get around the units I mean I tried assigning x to be gold and y silver but this $g/cm^3$  how do you equate that to $3823.6g$ and have the unit match up.
I mean all I have on my paper is $19.3x + 10.5y = 3823.6$  but I know that is wrong and where do I get the second equation.  I am truly stuck and frustrated.  my book doesn't have a solutions manual so I wouldn't know if it is right or wrong.  Please help me set this up
 A: Let $g$ be the volume of gold, in $ \text{cm}^3$ and $s$ be the volume of silver in the same units. The volume of the crown is $ 220 \space\text{cm}^3$, so
$$g +s = 220$$
Every $ \text{cm}^3$ of gold is $19.3\text{g}$ and the same volume of silver is  $10.5\text{g}$. Multiply the individual volume of gold & silver to get the total weight of the crown.
$$19.3g +10.5s = 3823.6$$
You now have 2 equations. Use simultaneous equations to solve the individual volume of gold & silver. Finally, find the percentage mass of gold by taking
$$\frac{19.3g}{3823.6g}$$
Update
You look like you're looking for the matrix way of solving. I'll re-write the equations for you in the form:
$$
\begin{align*}
g +s &= 220\\
19.3g +10.5s &= 3823.6
\end{align*}
$$
To put the equation in matrix form, 
$$
\begin{pmatrix}
1 & 1\\
19.3 & 10.5
\end{pmatrix}
\begin{pmatrix} g \\ s\end{pmatrix} =
\begin{pmatrix} 220 \\ 3823.6\end{pmatrix}
$$
Find the inverse of the matrix
$$
\begin{pmatrix}
1 & 1\\
19.3 & 10.5
\end{pmatrix}
$$
which is
$$
\begin{pmatrix}
-1.193 & 0.114\\
2.193 & 0-.114
\end{pmatrix}
$$
And the equation you want to solve will be
$$
\begin{pmatrix}
-1.193 & 0.114\\
2.193 & 0-.114
\end{pmatrix}
\begin{pmatrix}
1 & 1\\
19.3 & 10.5
\end{pmatrix}
\begin{pmatrix} g \\ s\end{pmatrix} =\begin{pmatrix}
-1.193 & 0.114\\
2.193 & 0-.114
\end{pmatrix}
\begin{pmatrix} 220 \\ 3823.6\end{pmatrix}
$$
The left hand side will evaluate to be a $2 \times 2$ identity matrix, which you have identified as $I$. The right hand side will give you the values of $g$ and $s$ respectively.
A: let the crown be entirely of gold. Then for every  $cm^3 $you make of silver instead of gold you will lose 8.8 grams off the total crown. if it was all made of gold it would weigh $19.3 \frac{grams}{cm^3}*220 cm^3 = 4246$ grams so therefore the crown weighs $4246-3823= 423$ grams less than the pure gold.
Since or every $cm^3$ you substitute with silver you reduce the weight by 8.8 grams then if x is the number of $cm^3$ of silver then $x*8.8=423$ so $x=\frac{423}{8.8} $which is roughly 48.07
Looking at it in a system of equations (slightly different method) it would look like this, where $x_1$ and $x_2$ are the $cm^3$ of gold and silver respectively: 
$x_1+x_2=220$ and $19.3x_1+10.5x_2=3823.6$ so now you just have to solve these two equalities. 
