1. We are given the system of simultaneous equations: $$y-2x=5$$ $$2y+x=0$$ 2. To use the elimination method, we will eliminate one variable by making the coefficients of that variable in both equations equal with opposite signs. 3. Multiply the first equation by 2 to match the coefficient of $y$ in the second equation: $$2(y-2x)=2(5)\Rightarrow 2y - 4x = 10$$ 4. The system is now: $$2y - 4x = 10$$ $$2y + x = 0$$ 5. Subtract the second equation from the first to eliminate $y$: $$(2y - 4x) - (2y + x) = 10 - 0$$ $$2y - 4x - 2y - x = 10$$ $$-5x = 10$$ 6. Solve for $x$: $$x = \frac{10}{-5} = -2$$ 7. Substitute $x = -2$ into one of the original equations, for example, $y - 2x = 5$: $$y - 2(-2) = 5$$ $$y + 4 = 5$$ $$y = 5 - 4 = 1$$ 8. The solution to the system is: $$x = -2, \quad y = 1$$ 9. Verify by substituting into the second equation $2y + x = 0$: $$2(1) + (-2) = 2 - 2 = 0$$ This confirms the solution is correct.