Answer:
Step-by-step explanation:To solve this problem, let's break it down step by step:
Step 1: Let's assume the number of first-place finishers is "x," the number of second-place finishers is "y," and the number of third-place finishers is "z."
Step 2: We are given that a first-place finish earns 5 points, a second-place finish earns 3 points, and a third-place finish earns 1 point. We know that there were 24 individual events in total and the team scored a total of 56 points.
Step 3: So, we can write the following equation based on the given information:
5x + 3y + 1z = 56
Step 4: We are also given that the number of third-place finishers is equal to the sum of the first-place and second-place finishers. Mathematically, this can be written as:
z = x + y
Step 5: Now, we have two equations:
5x + 3y + z = 56 (equation 1)
z = x + y (equation 2)
Step 6: To solve these equations, we can substitute equation 2 into equation 1:
5x + 3y + (x + y) = 56
Step 7: Simplifying the equation:
6x + 4y = 56
Step 8: We can divide the equation by 2 to make it simpler:
3x + 2y = 28
Step 9: Now, let's try to find possible values for "x" and "y" that satisfy this equation. One solution is:
x = 4
y = 5
Step 10: Now, substituting these values back into equation 2 to find "z":
z = 4 + 5
z = 9
Step 11: Therefore, there were 4 athletes who finished in first place, 5 athletes who finished in second place, and 9 athletes who finished in third place.
In summary, there were 4 athletes who finished in first place, 5 athletes who finished in second place, and 9 athletes who finished in third place at the county track and field championships.