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Math Help - This linear algebra question is killing me.

  1. #1
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    This linear algebra question is killing me.

    The question says:

    A zoo charges $6 for adults, $3 for students, and $.50 for children. One morning 79 people enter and pay a total of $207. Determine the possible numbers of adults, students, and children.


    So, I got the equations 6a+3s+0.5c = 207 and a+s+c=79. I then put them into a matrix and got it into REF that looks that

    [ 1 0 0.83333 -10]
    [ 0 1 1.83333 89]

    I feel this should be really easy, but i cannot get a solution with guess and check.
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  2. #2
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    Quote Originally Posted by complex123 View Post
    The question says:

    A zoo charges $6 for adults, $3 for students, and $.50 for children. One morning 79 people enter and pay a total of $207. Determine the possible numbers of adults, students, and children.


    So, I got the equations 6a+3s+0.5c = 207 and a+s+c=79. I then put them into a matrix and got it into REF that looks that

    [ 1 0 0.83333 -10]
    [ 0 1 1.83333 89]

    I feel this should be really easy, but i cannot get a solution with guess and check.
    You have two equations in three unknowns so, as a purely linear algebra problem, this has no single solutions. But here, of course, a, s, and c must be positive integers and the problem says "solutions", plural.

    However, your row-reduce matrix is wrong- you dropped a sign. (Notice that the top row is equivalent to the equation a+ .83333 c= -10. a and c cannot both be non-negative!)

    Your row reduced matrix should be \begin{bmatrix}1 & 0 & -\frac{5}{6} & -10 \\ 0 & 1  & \frac{11}{6} & 89\end{bmatrix}.

    That is equivalent to a- \frac{5}{6}c= -10 and s+ \frac{11}{6}= 89 or a= \frac{5}{6}c- 10 and s= 89- \frac{11}{6}c.

    Now, since a and s must be non-zero integers, it is clear that c must be a multiple of 6, at least 12, and no larger than 48 ( 89- \frac{11}{6}(48)= 1). That is, c can be 12, 18, 24, 30, 36, 42, or 48.

    If c= 12, a= 10-10= 0 and s= 89- 22= 67.
    If c= 18, a= 15- 10= 5 and s= 89- 33= 56, etc.
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