1. **State the problem:** Solve the system of linear equations: $$5(x + 1) - 2(y - 3) = 21$$ $$3(x - 2) + 4(y + 1) = 14$$ 2. **Expand and simplify each equation:** For the first equation: $$5x + 5 - 2y + 6 = 21$$ $$5x - 2y + 11 = 21$$ Subtract 11 from both sides: $$5x - 2y = 21 - 11$$ $$5x - 2y = 10$$ For the second equation: $$3x - 6 + 4y + 4 = 14$$ $$3x + 4y - 2 = 14$$ Add 2 to both sides: $$3x + 4y = 16$$ 3. **Write the simplified system:** $$5x - 2y = 10$$ $$3x + 4y = 16$$ 4. **Use the elimination method to solve:** Multiply the first equation by 2 to align coefficients of $y$: $$2(5x - 2y) = 2(10)$$ $$10x - 4y = 20$$ Now add this to the second equation: $$3x + 4y = 16$$ Adding: $$10x - 4y + 3x + 4y = 20 + 16$$ $$13x = 36$$ 5. **Solve for $x$:** $$x = \frac{36}{13}$$ 6. **Substitute $x$ back into one of the original simplified equations to find $y$:** Use $5x - 2y = 10$: $$5\left(\frac{36}{13}\right) - 2y = 10$$ $$\frac{180}{13} - 2y = 10$$ Subtract $\frac{180}{13}$ from both sides: $$-2y = 10 - \frac{180}{13}$$ Convert 10 to $\frac{130}{13}$: $$-2y = \frac{130}{13} - \frac{180}{13} = -\frac{50}{13}$$ Divide both sides by $-2$: $$y = \frac{-\frac{50}{13}}{-2} = \frac{50}{13 \times 2} = \frac{50}{26} = \frac{25}{13}$$ 7. **Final solution:** $$\boxed{x = \frac{36}{13}, \quad y = \frac{25}{13}}$$