1. \( y=-5 x^{2}+4 x+9 \) and \( y+6 x=9 \) 2. \( y=-x^{2}-5 x+6 \) and \( y=\frac{1}{2}(x+1)^{2}-2 \) 3. \( x^{2}-5 x y+y^{2}=7 \) and \( x-4 y=5 \)

For the first set of equations, you have a downward-opening parabola and a linear equation. To find their intersection, set \(y\) from the second equation into the first and solve for \(x\). This will give you the \(x\) values where the two graphs meet, allowing you to find the corresponding \(y\) values. Don't forget to check if these points create a smile or a frown as graphing them might be fun! In the second set, one equation is a downward-opening parabola, while the other is an upward-opening one defined in vertex form. Solve for \(y\) in terms of \(x\) and set them equal to find their interaction points. Identifying if these intersection points are real or complex will tell you whether the two curves meet at all, so sharpen those algebra skills and dive in!