1. **State the problem:** Solve the system of equations using the elimination method. 2. **First system:** $$2x - y = 17$$ $$x - y = 10$$ 3. **Eliminate $y$ by subtracting the second equation from the first:** $$ (2x - y) - (x - y) = 17 - 10 $$ $$ 2x - y - x + y = 7 $$ $$ x = 7 $$ 4. **Substitute $x=7$ into $x - y = 10$:** $$ 7 - y = 10 $$ $$ -y = 3 $$ $$ y = -3 $$ 5. **Solution for first system:** $x=7$, $y=-3$. --- 6. **Second system:** $$ -x - 7y = 9 $$ $$ -x + 9y = -23 $$ 7. **Eliminate $x$ by subtracting the first equation from the second:** $$ (-x + 9y) - (-x - 7y) = -23 - 9 $$ $$ -x + 9y + x + 7y = -32 $$ $$ 16y = -32 $$ $$ y = \frac{-32}{16} = -2 $$ 8. **Substitute $y=-2$ into $-x - 7y = 9$:** $$ -x - 7(-2) = 9 $$ $$ -x + 14 = 9 $$ $$ -x = -5 $$ $$ x = 5 $$ 9. **Solution for second system:** $x=5$, $y=-2$. --- 10. **Third system:** $$ -3x + y = -14 $$ $$ -2x - y = 9 $$ 11. **Add the two equations to eliminate $y$:** $$ (-3x + y) + (-2x - y) = -14 + 9 $$ $$ -3x + y - 2x - y = -5 $$ $$ -5x = -5 $$ $$ x = 1 $$ 12. **Substitute $x=1$ into $-3x + y = -14$:** $$ -3(1) + y = -14 $$ $$ -3 + y = -14 $$ $$ y = -11 $$ 13. **Solution for third system:** $x=1$, $y=-11$. --- 14. **Fourth system:** $$ 6x + 3y = 18 $$ $$ -x - 4y = -25 $$ 15. **Multiply second equation by 3 to align $x$ coefficients:** $$ 3(-x - 4y) = 3(-25) $$ $$ -3x - 12y = -75 $$ 16. **Add to first equation:** $$ (6x + 3y) + (-3x - 12y) = 18 + (-75) $$ $$ 3x - 9y = -57 $$ 17. **Divide entire equation by 3:** $$ \frac{3x - 9y}{3} = \frac{-57}{3} $$ $$ \cancel{3}x - \cancel{9}y = -19 $$ $$ x - 3y = -19 $$ 18. **Express $x$ in terms of $y$:** $$ x = -19 + 3y $$ 19. **Substitute into second original equation:** $$ -(-19 + 3y) - 4y = -25 $$ $$ 19 - 3y - 4y = -25 $$ $$ 19 - 7y = -25 $$ $$ -7y = -44 $$ $$ y = \frac{44}{7} $$ 20. **Substitute $y=\frac{44}{7}$ into $x = -19 + 3y$:** $$ x = -19 + 3 \times \frac{44}{7} = -19 + \frac{132}{7} = \frac{-133 + 132}{7} = \frac{-1}{7} $$ 21. **Solution for fourth system:** $x=\frac{-1}{7}$, $y=\frac{44}{7}$. --- 22. **Fifth system:** $$ x + 4y = -3 $$ $$ x + 7y = -12 $$ 23. **Subtract first equation from second:** $$ (x + 7y) - (x + 4y) = -12 - (-3) $$ $$ x + 7y - x - 4y = -9 $$ $$ 3y = -9 $$ $$ y = -3 $$ 24. **Substitute $y=-3$ into $x + 4y = -3$:** $$ x + 4(-3) = -3 $$ $$ x - 12 = -3 $$ $$ x = 9 $$ 25. **Solution for fifth system:** $x=9$, $y=-3$. --- 26. **Sixth system:** $$ 3x + y = -21 $$ $$ x + y = -5 $$ 27. **Subtract second equation from first:** $$ (3x + y) - (x + y) = -21 - (-5) $$ $$ 3x + y - x - y = -16 $$ $$ 2x = -16 $$ $$ x = -8 $$ 28. **Substitute $x=-8$ into $x + y = -5$:** $$ -8 + y = -5 $$ $$ y = 3 $$ 29. **Solution for sixth system:** $x=-8$, $y=3$. --- **Final answers:** 1) $x=7$, $y=-3$ 2) $x=5$, $y=-2$ 3) $x=1$, $y=-11$ 4) $x=\frac{-1}{7}$, $y=\frac{44}{7}$ 5) $x=9$, $y=-3$ 6) $x=-8$, $y=3$