sly \( 3 x+2=y \) and \( y=-x^{2}+2 x+8 \)

To solve the system of equations \( 3x + 2 = y \) and \( y = -x^2 + 2x + 8 \), we can set them equal to each other. This gives us: \[ 3x + 2 = -x^2 + 2x + 8 \] Rearranging this equation results in a quadratic equation: \[ x^2 + x - 6 = 0 \] Factoring the quadratic yields: \[ (x + 3)(x - 2) = 0 \] Thus, \( x = -3 \) or \( x = 2 \). Substituting these back into either equation provides the corresponding \( y \) values, allowing us to fully specify the intersection points of these curves! Now let's delve deeper into some of the nuances surrounding these equations: Solving systems of equations is like a dance between lines and curves! The linear equation \( y = 3x + 2 \) embodies the crispness of a straight line, while the quadratic \( y = -x^2 + 2x + 8 \) brings a sense of drama with its parabolic shape. When you superimpose these on a graph, you can visualize their intersection points as the magical spots where their paths cross—a true meeting of two worlds! For those eager to dabble in similar problems, grab a graphing calculator or an app! Plotting helps you catch insights about how these functions behave. Keep an eye out for points where they meet, as you'll be solving for variables in real-time, making math much more tangible and visually exciting. Plus, it’s a fantastic way to enhance your understanding of concepts like intercepts and symmetry!