1. **State the problem:** Solve the equation $7x - 5(x - 7) = 8x - 13$ for $x$. 2. **Apply the distributive property:** Expand $-5(x - 7)$ to get $-5x + 35$. 3. **Rewrite the equation:** $$7x - 5x + 35 = 8x - 13$$ 4. **Combine like terms on the left side:** $$2x + 35 = 8x - 13$$ 5. **Isolate variable terms on one side:** Subtract $2x$ from both sides: $$\cancel{2x} + 35 = 8x - \cancel{2x} - 13$$ which simplifies to $$35 = 6x - 13$$ 6. **Isolate the constant term:** Add $13$ to both sides: $$35 + 13 = 6x - 13 + 13$$ which simplifies to $$48 = 6x$$ 7. **Solve for $x$:** Divide both sides by $6$: $$\frac{48}{\cancel{6}} = \frac{6x}{\cancel{6}}$$ which simplifies to $$8 = x$$ **Final answer:** $$x = 8$$