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

Let’s dive into some cool background! The equation \( 2x^2 - 3xy = -4 \) is a quadratic equation in terms of \( x \) and represents a curve in a 2D plane. It can open up insights into geometry, like conic sections. The second equation, \( 4^{x+y} = 2^{y+4} \), shows how exponential functions can be manipulated using properties of exponents. Remember, \( 4 \) can be expressed as \( 2^2 \), which can pave the way for simplifying this equation. Now, let’s consider how these equations might pop up in the real world! You could see similar quadratic equations when modeling projectile motion or calculating areas. The second equation often surfaces in finance when dealing with compound interest or population growth problems, where exponential relationships take the stage. Connecting mathematical models to real situations helps visualize the fantastic utility of these concepts!