Answers:
- Ben = 6
- Max = 12
- Mitchell = 2
- Christion = 2
- Daniel = 18
Each count above refers to the number of baskets made for the given player.
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Explanation:
For each player, let the following variables represent how many baskets were made
- B = Ben
- M = Max
- T = Mitchel
- C = Christion
- D = Daniel
So for example, if D = 10, then this means Daniel made 10 baskets. Though for now, we're not sure what number replaces D or any given letter.
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"Daniel remembers that he made three times as many baskets as Ben" tells us that D = 3B is one equation
We know another equation is M = B+6 since Max made six more baskets than Ben
Another equation is C = T because these two students (Christion and Mitchell) made the same number of baskets.
We also know that T < B since Mitchell made fewer baskets compared to Ben.
Altogether the boys made 40 baskets total, so B+M+T+C+D = 40
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We have these five mathematical statements
- D = 3B
- M = B+6
- C = T
- T < B
- B+M+T+C+D = 40
Let's focus on the last equation
B+M+T+C+D = 40
B+M+T+C+3B = 40 .... plug in D = 3B
4B+M+T+C = 40
4B+M+T+T = 40 .... plug in C = T
4B+M+2T = 40
4B+B+6+2T = 40 .... plug in M = B+6
5B+6+2T = 40
5B+2T = 40-6
5B+2T = 34
Now let's solve for T
5B+2T = 34
2T = 34-5B
2T = -5B+34
T = (-5B+34)/2
T = (-5B)/2 + 34/2
T = -2.5B + 17
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We've eliminated everything but two variables at this point.
Let's say that x replaced B and y replaces T
We get the equation y = -2.5x + 17
If you graph this equation along with y = x, you'll find that the two equations intersect at roughly (4.857, 4.857)
Since we want T < B, this means we want y < x to be the case. The line y = -2.5x+17 goes downhill as we move from left to right. If x is smaller than 4.857, then x >= y. Otherwise, if x is larger than 4.857, then y < x
Furthermore, looking at a table of values for this equation shows that only (x,y) = (6,2) is a point on y = -2.5x+17 such that both x and y are positive and x is the larger value, and they are both integer coordinates.
All of this means that when B = 6, this pairs with T = 2. No other number pair is possible for these two students.
We then use this to find the other player's basket count
- D = 3B = 3*6 = 18
- M = B+6 = 6+6 = 12
- C = T = 2
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So we have the following results
- B = 6
- M = 12
- T = 2
- C = 2
- D = 18
As a check,
B+M+T+C+D = 6+12+2+2+18 = 40
which confirms our answers.
