Can you find the mass of solvent with mass of solute, volume of solution, and solution density? If you have, for example, $\pu{10 g}$ of a solute dissolved into $\pu{100 ml}$ of a solution, and the solution density is $\pu{1.5 g ml^-1}$, can you just multiply $100\ \mathrm{ml} \times 1.5\ \mathrm{\frac{g}{ml}}$, then subtract the $\pu{10 g}$ of solute from the result to get the mass of the solvent in grams? 

Does that work or am I trying to do do this wrong?
That works!  In most chemistry (but not nuclear reactions!),  mass is always conserved, so 

$$m_\text{solution} = m_\text{solvent} + m_\text{solutes}$$

Mass is also equal to density times volume, so

$$m_\text{solution} = \rho_\text{solution} V_\text{solution} = m_\text{solvent} + m_\text{solutes}$$

Solving for $m_\text{solvent}$:

$$m_\text{solvent} = \rho_\text{solution} V_\text{solution} - m_\text{solutes}$$

Which is exactly the calculation you did.