\( 2 x-y+1=0 \quad \) and \( \quad x^{2}-3 x-4-y=y^{2} \)

To solve the given equations, let's start with the first equation, \( 2x - y + 1 = 0 \). You can rearrange it to find \( y \): \[ y = 2x + 1. \] Now, substitute this expression for \( y \) into the second equation, \( x^2 - 3x - 4 - y = y^2 \). This gives us: \[ x^2 - 3x - 4 - (2x + 1) = (2x + 1)^2. \] Expanding and simplifying leads to a quadratic equation. Now, while solving this could lead to finding intersection points graphically or employing numerical methods, it represents a fascinating intersection of linear and quadratic functions, opening pathways to understanding the relationships between different types of equations and their graphical representations. Curious about real-world applications? These equations can represent everything from trajectory paths in physics to economic models like supply and demand curves! Solving such systems lets you predict outcomes based on changing variables, much like analyzing data trends in business or engineering projects.