1. **Problem:** Multiply the equation $2x - 3y = 5$ by 2. Formula: Multiply each term by 2. Work: $$2 \times (2x - 3y) = 2 \times 5$$ $$4x - 6y = 10$$ 2. **Problem:** Multiply the equation $x + 3y = 7$ by -3. Work: $$-3 \times (x + 3y) = -3 \times 7$$ $$-3x - 9y = -21$$ 3. **Problem:** Multiply the equation $2x + 5y = 1$ by 4. Work: $$4 \times (2x + 5y) = 4 \times 1$$ $$8x + 20y = 4$$ 4. **Problem:** Multiply the equation $3x - 2y = 8$ by -2. Work: $$-2 \times (3x - 2y) = -2 \times 8$$ $$-6x + 4y = -16$$ 5. **Problem:** Multiply the equation $5x - y = 2$ by 5. Work: $$5 \times (5x - y) = 5 \times 2$$ $$25x - 5y = 10$$ 6. **Problem:** Multiply the equation $-2x + 5y = -1$ by -1. Work: $$-1 \times (-2x + 5y) = -1 \times (-1)$$ $$2x - 5y = 1$$ --- 7. **Problem:** Add vertically the equations $3x + 2y = 6$ and $x - 2y = 10$. Work: $$(3x + 2y) + (x - 2y) = 6 + 10$$ $$3x + 2y + x - 2y = 16$$ $$4x = 16$$ 8. **Problem:** Add vertically the equations $3x - y = 8$ and $2x + y = 7$. Work: $$(3x - y) + (2x + y) = 8 + 7$$ $$3x - y + 2x + y = 15$$ $$5x = 15$$ 9. **Problem:** Add vertically the equations $x + y = 5$ and $x - y = 7$. Work: $$(x + y) + (x - y) = 5 + 7$$ $$x + y + x - y = 12$$ $$2x = 12$$ 10. **Problem:** Add vertically the equations $3x - y = 4$ and $-3x + 4y = 2$. Work: $$(3x - y) + (-3x + 4y) = 4 + 2$$ $$3x - y - 3x + 4y = 6$$ $$3y = 6$$ 11. **Problem:** Add vertically the equations $5x - y = 6$ and $-5x + 3y = -8$. Work: $$(5x - y) + (-5x + 3y) = 6 + (-8)$$ $$5x - y - 5x + 3y = -2$$ $$2y = -2$$ 12. **Problem:** Add vertically the equations $-8x + 2y = 11$ and $8x - 3y = -7$. Work: $$(-8x + 2y) + (8x - 3y) = 11 + (-7)$$ $$-8x + 2y + 8x - 3y = 4$$ $$-y = 4$$ --- 13. **Problem:** Solve simultaneously using elimination: $3x + y = 13$ and $x - y = 3$. Add equations: $$(3x + y) + (x - y) = 13 + 3$$ $$4x = 16$$ $$x = 4$$ Substitute $x=4$ into $x - y = 3$: $$4 - y = 3$$ $$y = 1$$ 14. **Problem:** Solve $2x - y = 8$ and $3x + y = 7$. Add equations: $$(2x - y) + (3x + y) = 8 + 7$$ $$5x = 15$$ $$x = 3$$ Substitute $x=3$ into $2x - y = 8$: $$2(3) - y = 8$$ $$6 - y = 8$$ $$y = -2$$ 15. **Problem:** Solve $x + 3y = 13$ and $-x + y = 7$. Add equations: $$(x + 3y) + (-x + y) = 13 + 7$$ $$4y = 20$$ $$y = 5$$ Substitute $y=5$ into $x + 3y = 13$: $$x + 3(5) = 13$$ $$x + 15 = 13$$ $$x = -2$$ 16. **Problem:** Solve $5x + 2y = -19$ and $3x - 4y = -1$. Multiply first by 2 and second by 1 to eliminate $y$: $$2(5x + 2y) = 2(-19)$$ $$10x + 4y = -38$$ Add to second: $$(10x + 4y) + (3x - 4y) = -38 + (-1)$$ $$13x = -39$$ $$x = -3$$ Substitute $x=-3$ into $5x + 2y = -19$: $$5(-3) + 2y = -19$$ $$-15 + 2y = -19$$ $$2y = -4$$ $$y = -2$$ 17. **Problem:** Solve $2x + 3y = 11$ and $7x - y = 50$. Multiply second by 3: $$3(7x - y) = 3(50)$$ $$21x - 3y = 150$$ Add to first: $$(2x + 3y) + (21x - 3y) = 11 + 150$$ $$23x = 161$$ $$x = \frac{161}{23} = 7$$ Substitute $x=7$ into $2x + 3y = 11$: $$2(7) + 3y = 11$$ $$14 + 3y = 11$$ $$3y = -3$$ $$y = -1$$ 18. **Problem:** Solve $2x + y = 1$ and $x + 3y = -12$. Multiply second by 2: $$2(x + 3y) = 2(-12)$$ $$2x + 6y = -24$$ Subtract first from this: $$(2x + 6y) - (2x + y) = -24 - 1$$ $$5y = -25$$ $$y = -5$$ Substitute $y=-5$ into $2x + y = 1$: $$2x - 5 = 1$$ $$2x = 6$$ $$x = 3$$ 19. **Problem:** Solve $4x + y = 19$ and $3x + 4y = -2$. Multiply first by 4: $$4(4x + y) = 4(19)$$ $$16x + 4y = 76$$ Subtract second: $$(16x + 4y) - (3x + 4y) = 76 - (-2)$$ $$13x = 78$$ $$x = 6$$ Substitute $x=6$ into $4x + y = 19$: $$4(6) + y = 19$$ $$24 + y = 19$$ $$y = -5$$ 20. **Problem:** Solve $7x + 2y = -5$ and $3x - 5y = -49$. Multiply first by 5 and second by 2: $$5(7x + 2y) = 5(-5)$$ $$35x + 10y = -25$$ $$2(3x - 5y) = 2(-49)$$ $$6x - 10y = -98$$ Add: $$(35x + 10y) + (6x - 10y) = -25 + (-98)$$ $$41x = -123$$ $$x = -3$$ Substitute $x=-3$ into $7x + 2y = -5$: $$7(-3) + 2y = -5$$ $$-21 + 2y = -5$$ $$2y = 16$$ $$y = 8$$ 21. **Problem:** Solve $6x + 5y = -2$ and $3x - y = 13$. Multiply second by 5: $$5(3x - y) = 5(13)$$ $$15x - 5y = 65$$ Add to first: $$(6x + 5y) + (15x - 5y) = -2 + 65$$ $$21x = 63$$ $$x = 3$$ Substitute $x=3$ into $3x - y = 13$: $$3(3) - y = 13$$ $$9 - y = 13$$ $$y = -4$$ 22. **Problem:** Solve $4x - 3y = 12$ and $-x + 5y = -3$. Multiply second by 4: $$4(-x + 5y) = 4(-3)$$ $$-4x + 20y = -12$$ Add to first: $$(4x - 3y) + (-4x + 20y) = 12 + (-12)$$ $$17y = 0$$ $$y = 0$$ Substitute $y=0$ into $4x - 3y = 12$: $$4x = 12$$ $$x = 3$$ 23. **Problem:** Solve $3x + 2y = 7$ and $8x + 7y = 12$. Multiply first by 7 and second by -2: $$7(3x + 2y) = 7(7)$$ $$21x + 14y = 49$$ $$-2(8x + 7y) = -2(12)$$ $$-16x - 14y = -24$$ Add: $$(21x + 14y) + (-16x - 14y) = 49 + (-24)$$ $$5x = 25$$ $$x = 5$$ Substitute $x=5$ into $3x + 2y = 7$: $$3(5) + 2y = 7$$ $$15 + 2y = 7$$ $$2y = -8$$ $$y = -4$$ 24. **Problem:** Solve $3x + 7y = 47$ and $7x + 3y = 43$. Multiply first by 3 and second by -7: $$3(3x + 7y) = 3(47)$$ $$9x + 21y = 141$$ $$-7(7x + 3y) = -7(43)$$ $$-49x - 21y = -301$$ Add: $$(9x + 21y) + (-49x - 21y) = 141 + (-301)$$ $$-40x = -160$$ $$x = 4$$ Substitute $x=4$ into $3x + 7y = 47$: $$3(4) + 7y = 47$$ $$12 + 7y = 47$$ $$7y = 35$$ $$y = 5$$ 25. **Problem:** Solve $2x + 7y = -51$ and $3x - 2y = 11$. Multiply first by 2 and second by 7: $$2(2x + 7y) = 2(-51)$$ $$4x + 14y = -102$$ $$7(3x - 2y) = 7(11)$$ $$21x - 14y = 77$$ Add: $$(4x + 14y) + (21x - 14y) = -102 + 77$$ $$25x = -25$$ $$x = -1$$ Substitute $x=-1$ into $2x + 7y = -51$: $$2(-1) + 7y = -51$$ $$-2 + 7y = -51$$ $$7y = -49$$ $$y = -7$$ 26. **Problem:** Solve $3x + y = 17$ and $2x - y = 23$. Add equations: $$(3x + y) + (2x - y) = 17 + 23$$ $$5x = 40$$ $$x = 8$$ Substitute $x=8$ into $3x + y = 17$: $$3(8) + y = 17$$ $$24 + y = 17$$ $$y = -7$$ 27. **Problem:** Solve $2x - 3y = 14$ and $5x - 7y = 34$. Multiply first by 5 and second by -2: $$5(2x - 3y) = 5(14)$$ $$10x - 15y = 70$$ $$-2(5x - 7y) = -2(34)$$ $$-10x + 14y = -68$$ Add: $$(10x - 15y) + (-10x + 14y) = 70 + (-68)$$ $$-y = 2$$ $$y = -2$$ Substitute $y=-2$ into $2x - 3y = 14$: $$2x - 3(-2) = 14$$ $$2x + 6 = 14$$ $$2x = 8$$ $$x = 4$$