1. **Stating the problem:** Solve the system of equations using the mixed method (Elimination-Substitution): $$\begin{cases} 3x - 5y = 5 \\ 4x - 3y = 14 \end{cases}$$ 2. **Formula and rules:** - Elimination method involves adding or subtracting equations to eliminate one variable. - Substitution method involves solving one equation for one variable and substituting into the other. 3. **Step 1: Eliminate one variable.** Multiply the first equation by 3 and the second by 5 to align coefficients of $y$: $$\begin{cases} 3(3x - 5y) = 3(5) \\ 5(4x - 3y) = 5(14) \end{cases}$$ which gives $$\begin{cases} 9x - 15y = 15 \\ 20x - 15y = 70 \end{cases}$$ 4. **Step 2: Subtract the first from the second to eliminate $y$:** $$ (20x - 15y) - (9x - 15y) = 70 - 15 $$ $$ 20x - 15y - 9x + 15y = 55 $$ $$ (20x - 9x) + (-15y + 15y) = 55 $$ $$ 11x + 0 = 55 $$ $$ 11x = 55 $$ 5. **Step 3: Solve for $x$:** $$ x = \frac{55}{11} $$ Intermediate step showing cancellation: $$ x = \frac{\cancel{55}}{\cancel{11}} = 5 $$ 6. **Step 4: Substitute $x=5$ into the first original equation to find $y$:** $$ 3(5) - 5y = 5 $$ $$ 15 - 5y = 5 $$ 7. **Step 5: Solve for $y$:** $$ -5y = 5 - 15 $$ $$ -5y = -10 $$ $$ y = \frac{-10}{-5} $$ Intermediate step showing cancellation: $$ y = \frac{\cancel{-10}}{\cancel{-5}} = 2 $$ 8. **Final answer:** $$ \boxed{x=5, y=2} $$