1. **State the problem:** We are given two equations: $$5x - 2y = -4$$ $$5x^2 - 4y^2 = -16$$ We want to analyze and understand these equations. 2. **Identify the types of equations:** - The first equation is linear in $x$ and $y$. - The second equation is quadratic and resembles a conic section. 3. **Rewrite the first equation to express $y$ in terms of $x$:** $$5x - 2y = -4$$ Subtract $5x$ from both sides: $$-2y = -4 - 5x$$ Divide both sides by $-2$: $$y = \frac{-4 - 5x}{-2} = \frac{\cancel{-4} - 5x}{\cancel{-2}} = 2 + \frac{5}{2}x$$ So, $$y = 2 + \frac{5}{2}x$$ 4. **Rewrite the second equation:** $$5x^2 - 4y^2 = -16$$ Add $16$ to both sides: $$5x^2 - 4y^2 + 16 = 0$$ Or equivalently: $$5x^2 - 4y^2 = -16$$ Divide both sides by $-16$: $$\frac{5x^2}{-16} - \frac{4y^2}{-16} = 1$$ Simplify: $$-\frac{5}{16}x^2 + \frac{1}{4}y^2 = 1$$ Multiply both sides by $-1$: $$\frac{5}{16}x^2 - \frac{1}{4}y^2 = -1$$ This is a hyperbola equation in standard form. 5. **Summary:** - The first equation is a line: $y = 2 + \frac{5}{2}x$. - The second equation is a hyperbola: $5x^2 - 4y^2 = -16$. These two equations describe a system with a linear and a quadratic curve.