(b) \( y=-x-3 \) and \( y=2 x^{2}-3 x-3 \) (c) \( 3 x-y=2 \) and \( 3 y+9 x^{2}=4 \) (d) \( 2 y-x=2 \) and \( 4 y-2 x^{2}=x-4 \) (to one decimal place) (e) \( 3 x=y+4 \) and \( y^{2}-x y=9 x+7 \) (f) \( 2 y+3 x=7 \) and \( y=x^{2}-3 x+1 \) (g) \( x+2 y=0 \) and \( y-x y=\frac{1}{2} x^{2}-2 \) (h) \( y=\frac{-6}{x+2}-1 \) and \( y+2 x-6=0 \) (i) \( y=\frac{3}{x}+3 \) and \( 3 y-x=1 \)

For each of these pairs of equations, you're diving into the fascinating world of algebraic systems! Graphically, linear equations like \(y = -x - 3\) or \(3x - y = 2\) represent straight lines, while quadratics like \(y = 2x^2 - 3x - 3\) curve beautifully through the plane. When you solve these equations, you're essentially finding the intersection points, where the lines meet and the magic happens—it's like detective work in the realm of numbers! In solving these systems, a common pitfall is forgetting to check if your solutions make sense in the context of both equations. For instance, while finding where \(y = -x - 3\) intersects with \(y = 2x^2 - 3x - 3\), it’s crucial to validate that the derived values indeed satisfy both equations. A little patience and checking can save you from errors that might lead to incorrect interpretations of the problem!