1. **State the problem**: We are given two simultaneous equations: $$4x^2 - y^2 = 15$$ $$2x - y = 5$$ Our goal is to find the values of $x$ and $y$ that satisfy both equations. 2. **Express $y$ from the second equation**: From $$2x - y = 5$$, $$y = 2x - 5$$ 3. **Substitute $y$ into the first equation**: Replace $y$ in $$4x^2 - y^2 = 15$$ with $2x - 5$: $$4x^2 - (2x - 5)^2 = 15$$ 4. **Expand and simplify**: $$4x^2 - (4x^2 - 20x + 25) = 15$$ $$4x^2 - 4x^2 + 20x - 25 = 15$$ $$20x - 25 = 15$$ 5. **Solve for $x$**: $$20x = 15 + 25$$ $$20x = 40$$ $$x = \frac{40}{20} = 2$$ 6. **Find $y$ using $x=2$**: Substitute into $$y = 2x - 5$$: $$y = 2(2) - 5 = 4 - 5 = -1$$ 7. **Verify solution**: Plug $x=2$ and $y=-1$ into the first equation: $$4(2)^2 - (-1)^2 = 4(4) - 1 = 16 - 1 = 15$$ which is true. **Final answer:** $$\boxed{x=2, y=-1}$$