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Math Help - 3x-14y=2 (solving over integers)

  1. #1
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    3x-14y=2 (solving over integers)

    Can somebody please show me how you are supposed to solve this over integers?

    3x - 14y = 2

    This is what WolframAlpha yields, but it doesn't show any steps.
    solve over integers 3x-14y=2 - Wolfram|Alpha
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  2. #2
    Member anonimnystefy's Avatar
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    Re: 3x-14y=2 (solving over integers)

    hi

    You should be able to do it using the Extended Euclidean Algorithm shown at Extended Euclidean algorithm - Wikipedia, the free encyclopedia

    If you have more questions post here.

    Stefy
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  3. #3
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    Re: 3x-14y=2 (solving over integers)

    Hello, SweatingBear!

    Here is an approaching using only Algebra.


    Can somebody please show me how you are supposed to solve this over integers?

    . . 3x - 14y \:=\: 2

    Solve for x\!:\;\;3x \:=\:14y + 2 \qud\Rightarrow\quad x \:=\:\frac{14y+2}{3}

    So we have: . x \:=\:4y + \frac{2y+2}{3} .[1]

    Since x is an integer, 2y+2 must be a multiple of 3.
    . . 2y + 2 \:=\:3a\:\text{ for some integer }a.

    Then: . y \:=\:\frac{3a-2}{2} \quad\Rightarrow\quad y\:=\:a + \frac{a-2}{2} .[2]

    Since y is an integer, a-2 must be a multiple of 2.
    . . a - 2 \:=\:2b \quad\Rightarrow\quad a \:=\:2b+2


    Substitute into [2]: . y \:=\:(2b+2) + \frac{(2b+2)-2}{2} \quad\Rightarrow\quad y \:=\:3b+2

    Substitute into [1]: . x \:=\:4(3b+2) + \frac{2(3b+2)+2}{3} \quad\Rightarrow\quad x \:=\:14b + 10


    The solutions are: . \begin{Bmatrix}x &=& 14b + 10 \\ y &=& 3b + 2\end{Bmatrix}\;\text{ for any integer }b.
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