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A school sells two types of tickets for a play: adult tickets and student tickets. Adult tickets are `\$5` each and student tickets are `\$2` each. The school collects `\$400` total.
Here is your equation from the previous screen: `5x\ +\ 2y\ =\ 400`
How many of each ticket type could the school sell to collect `\$400`?
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GPT-5 mini
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Solve 5x + 2y = 400 for nonnegative integers x (adults) and y (students).
Rearrange: y = (400 − 5x)/2. For y to be an integer, 400 − 5x must be even, so x must be even. Let x = 2k. Then
y = 200 − 5k, with k an integer and 0 ≤ k ≤ 40 (so y ≥ 0).
Thus all solutions are
x = 2k, y = 200 − 5k for k = 0, 1, 2, …, 40.
Examples: (x,y) = (0,200), (2,195), (4,190), …, (80,0). There are 41 possible nonnegative solutions. If you require both types sold (x>0 and y>0), use k = 1..39 (39 solutions).
Rearrange: y = (400 − 5x)/2. For y to be an integer, 400 − 5x must be even, so x must be even. Let x = 2k. Then
y = 200 − 5k, with k an integer and 0 ≤ k ≤ 40 (so y ≥ 0).
Thus all solutions are
x = 2k, y = 200 − 5k for k = 0, 1, 2, …, 40.
Examples: (x,y) = (0,200), (2,195), (4,190), …, (80,0). There are 41 possible nonnegative solutions. If you require both types sold (x>0 and y>0), use k = 1..39 (39 solutions).
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