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Math Help - possible values for simple equation

  1. #1
    Senior Member DivideBy0's Avatar
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    possible values for simple equation

    4m+ 3n= 400

    Express all possible positive integer values of m, n in terms of a third variable, k. (Use modular arithmetic)
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  2. #2
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    Hello, DivideBy0!

    We can solve this with "normal" algebra . . .


    4m+ 3n\:=\: 400

    Express all possible positive integer values of m,\, n in terms of a third variable, k.

    Solve for m\!:\;\;m \:=\;100-\frac{3}{4}n .[1]

    Since m is an integer, n must be a multiple of 4.
    . . That is: . n \:=\:4k for some integer k.

    Substitue into [1]: . m \:=\:100-\frac{3}{4}(4k)\:=\:100-3k

    And we have parametric equations for all solutions:

    . . \begin{array}{ccc}m & = & 100-3k \\ n & = & 4k\end{array} . for any integer k.

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  3. #3
    Senior Member DivideBy0's Avatar
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    Thanks, I have two question though
    How would you express them if you could only have them as natural numbers, would you have 1 =< k =< 33?
    and What are parametric equations? thanks again
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by DivideBy0 View Post
    Thanks, I have two question though
    How would you express them if you could only have them as natural numbers, would you have 1 =< k =< 33?
    and What are parametric equations? thanks again
    As

    <br />
4m+3n=400<br />

    4|n, so n=4k for some k \in \bold{Z}

    Then we have:

    <br />
m+k=100<br />
,

    or:

    <br />
m=100-k<br />
.

    Now if you choose any k \in \bold{Z} then:

    m=100-k,\ n=4k

    is a solution to the original equation.

    RonL
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  5. #5
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    Hello, DivideBy0!

    How would you express them if you could only have them as natural numbers?
    Would you have: . 1 \:\leq\: k \:\leq \:33? . . . . Yes!

    What are parametric equations?

    A parameter is an "extra variable".

    Instead of having y as a function of x\!:\;y \:=\:f(x),

    . . we can have: . \begin{array}{cc}x\text{ as a function of }t\!: & x \:=\:g(t) \\<br />
y\text{ as a function of }t\!: & y \:=\:h(t)\end{array}

    This opens the door to an entire universe of fascinating functions and graphs
    . . . . . curves with loops, that intersect itself, etc.

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