1. **State the problem:** Solve the system of equations: $$7x - 5y = 13$$ $$9y - 4x = 2(x + y) + 1$$ 2. **Rewrite the second equation:** Expand the right side: $$9y - 4x = 2x + 2y + 1$$ Bring all terms to one side: $$9y - 4x - 2x - 2y = 1$$ Simplify: $$7y - 6x = 1$$ 3. **System to solve:** $$7x - 5y = 13$$ $$-6x + 7y = 1$$ 4. **Use elimination method:** Multiply the first equation by 7 and the second by 5 to align coefficients of $y$: $$7(7x - 5y) = 7(13) \Rightarrow 49x - 35y = 91$$ $$5(-6x + 7y) = 5(1) \Rightarrow -30x + 35y = 5$$ 5. **Add the two equations:** $$(49x - 35y) + (-30x + 35y) = 91 + 5$$ $$19x = 96$$ 6. **Solve for $x$:** $$x = \frac{96}{19}$$ 7. **Substitute $x$ back into the first original equation:** $$7\left(\frac{96}{19}\right) - 5y = 13$$ $$\frac{672}{19} - 5y = 13$$ 8. **Isolate $y$:** $$-5y = 13 - \frac{672}{19} = \frac{247}{19} - \frac{672}{19} = -\frac{425}{19}$$ $$y = \frac{425}{19 \times 5} = \frac{425}{95} = \frac{85}{19}$$ **Final answer:** $$x = \frac{96}{19}, \quad y = \frac{85}{19}$$