Which ordered pair is a solution \( y=x^{2}-6 x+11 \) \( y=-3 x+9 \)

Let's find the intersection of the two equations by setting them equal to each other since both are equal to \( y \): \( x^{2} - 6x + 11 = -3x + 9 \). Rearranging gives: \( x^{2} - 3x + 2 = 0 \). Factoring this gives: \( (x - 1)(x - 2) = 0 \). So, \( x = 1 \) and \( x = 2 \). Plugging both values back into either equation (let’s use \( y = -3x + 9 \)): For \( x = 1 \): \( y = -3(1) + 9 = 6 \). So, one solution is \( (1, 6) \). For \( x = 2 \): \( y = -3(2) + 9 = 3 \). So, another solution is \( (2, 3) \). Therefore, the ordered pairs that are solutions are \( (1, 6) \) and \( (2, 3) \). Now, the next logical step is figuring out how we can analyze the graph of both equations to better understand their intersection. Graphing the first equation will give you a parabola opening upwards, while the second will yield a straight line. Their points of intersection represent the pairs we found—like solving a mystery of where two paths cross! Moreover, remember that substituting values back into the original equations is crucial when solving such systems, as this helps confirm your solutions. Always check your work to avoid mistakes, like miscalculation or overlooking a possible pair!