1. **Stating the problem:** Solve the systems of equations by isolating one variable without generating fractions. 2. **System 1:** $$\begin{cases} x + y = 7 \\ x - y = 1 \end{cases}$$ 3. **Isolate $x$ in the first equation:** $$x = 7 - y$$ 4. **Substitute $x$ into the second equation:** $$ (7 - y) - y = 1 $$ 5. **Simplify:** $$ 7 - 2y = 1 $$ 6. **Isolate $y$:** $$ 7 - 2y = 1 \implies -2y = 1 - 7 \implies -2y = -6 $$ 7. **Divide both sides by $-2$ (showing cancellation):** $$ y = \frac{-6}{-2} = \frac{\cancel{-6}}{\cancel{-2}} = 3 $$ 8. **Find $x$ using $x = 7 - y$:** $$ x = 7 - 3 = 4 $$ 9. **Solution for system 1:** $$ (x, y) = (4, 3) $$ --- 10. **System 2:** $$\begin{cases} 2x + y = 10 \\ x + y = 7 \end{cases}$$ 11. **Isolate $y$ in the second equation:** $$ y = 7 - x $$ 12. **Substitute $y$ into the first equation:** $$ 2x + (7 - x) = 10 $$ 13. **Simplify:** $$ 2x + 7 - x = 10 \implies x + 7 = 10 $$ 14. **Isolate $x$:** $$ x = 10 - 7 = 3 $$ 15. **Find $y$ using $y = 7 - x$:** $$ y = 7 - 3 = 4 $$ 16. **Solution for system 2:** $$ (x, y) = (3, 4) $$