1. **State the problem:** Solve the system of equations: $$3x - 4y = 10$$ $$5x - 8y = 22$$ 2. **Choose a method:** We will use the elimination method to solve for $x$ and $y$. 3. **Eliminate one variable:** Multiply the first equation by 2 to align the coefficients of $y$: $$2(3x - 4y) = 2(10)$$ $$6x - 8y = 20$$ 4. **Subtract the second equation from this new equation:** $$\cancel{6x} - 8y - (5x - 8y) = 20 - 22$$ $$6x - 8y - 5x + 8y = -2$$ $$x = -2$$ 5. **Substitute $x = -2$ into the first original equation:** $$3(-2) - 4y = 10$$ $$-6 - 4y = 10$$ 6. **Solve for $y$:** $$-4y = 10 + 6$$ $$-4y = 16$$ $$y = \frac{16}{-4} = -4$$ 7. **Final answer:** $$x = -2, \quad y = -4$$