9. In a basketball garne, Marlene made 16 field goals. Each of the field goals was worth either 2 points or 3 points, and Marlene scored a total of 39 points from field goals. Let \( x \) represent the number of 2 point field goals and \( y \) represent the number of 3 point field goals. - Write a systern of equations in terms of \( x \) and \( y \) to model the situation. - How many 2-point field goals and how many 3-point field goals did Marlene make? - Explain or show how can you verify your solution by checking both the number of field goals and the total points scored? \( \square \) OV 50000

To solve the problem, we can set up a system of equations based on the information given. First, we know that Marlene made a total of 16 field goals, which we can express as: \[ x + y = 16 \] (Equation 1: the total number of field goals). Second, since each 2-point field goal contributes 2 points and each 3-point field goal contributes 3 points, we have: \[ 2x + 3y = 39 \] (Equation 2: the total points scored). Now we can solve this system of equations! To solve for \( x \) and \( y \), we can express \( y \) in Equation 1: \[ y = 16 - x \] Substituting \( y \) in Equation 2 gives: \[ 2x + 3(16 - x) = 39 \] This simplifies to: \[ 2x + 48 - 3x = 39 \] \[ -x + 48 = 39 \] \[ -x = 39 - 48 \] \[ -x = -9 \] \[ x = 9 \] Now plug \( x \) back into Equation 1 to find \( y \): \[ 9 + y = 16 \] \[ y = 16 - 9 \] \[ y = 7 \] So Marlene made 9 two-point field goals and 7 three-point field goals. To verify our solution, we can check both equations. Adding \( 9 \) (2-point shots) and \( 7 \) (3-point shots) gives: \[ 9 + 7 = 16 \quad (correct) \] Calculating the total points: \[ 2(9) + 3(7) = 18 + 21 = 39 \quad (correct) \] Both of the equations are satisfied, confirming our results! In the world of sports, statistics play an essential role, helping coaches strategize and players refine their skills. Basketball teams often analyze player performance data to identify patterns and enhance game strategies. Understanding scoring converts, like Marlene's field goals, can aid in developing strong offensive tactics, optimizing training regimens, and enhancing team communication. If you're curious about diving deeper into math applications in sports, there are loads of interesting resources out there! Books like "Moneyball" by Michael Lewis explore how statistics transformed baseball, but the principles extend to basketball too! Online courses focusing on sports analytics delve into practical examples involving stats, player performance metrics, and even fantasy sports strategies. Get ready to elevate your game!