Solve the system of equations: \begin{cases} 4x + 3y = 15, \\ 3x - y = 8. \end{cases}

Solving the system of equations:

We have the following system of equations:

\[\begin{cases} 4x + 3y = 15, \\ 3x - y = 8. \end{cases}\]

We can solve this system using substitution or elimination. Let's use elimination.

Step-by-step solution:

1. Multiply the second equation by 3 to eliminate y:
\[3(3x - y) = 3(8) \Rightarrow 9x - 3y = 24\]
2. Now add the modified second equation to the first equation:
\[(4x + 3y) + (9x - 3y) = 15 + 24 \Rightarrow 13x = 39\]
3. Divide by 13 to solve for x:
\[x = \frac{39}{13} = 3\]
4. Substitute the value of x into the second original equation to solve for y:
\[3(3) - y = 8 \Rightarrow 9 - y = 8 \Rightarrow y = 9 - 8 = 1\]

Answer: x = 3, y = 1