The referee at a college football game is standing 12 yards behind the ball. The back judge is standing 22 yards in front of the ball. The field judge is standing exactly halfway between them. Using coordinates to model the situation, find how far in front of the ball the field judge is. Explain.

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Establish Coordinate System: Let's establish a coordinate system where the position of the ball is at 0. The referee is at -12 yards (since he is behind the ball), and the back judge is at +22 yards (since he is in front of the ball).Calculate Distance Between Referee and Back Judge: The distance between the referee and the back judge is the sum of their distances from the ball, which is 12 yards + 22 yards = 34 yards.Determine Field Judge's Position: Since the field judge is exactly halfway between the referee and the back judge, the field judge is standing at half the distance of 34 yards from the referee's position. Half of 34 yards is 34 / 2 = 17 yards.Calculate Field Judge's Position Relative to Ball: To find the field judge's position relative to the ball, we add the referee's position to the halfway distance. The referee is at -12 yards, so we add 17 yards to this position. -12 yards + 17 yards = 5 yards.Therefore, the field judge is standing 5 yards in front of the ball.

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