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Math Help - Help Solving Linear Equations with fractions

  1. #1
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    Help Solving Linear Equations with fractions

    Okay, I am stuck on linear equations again. I am not understanding what to do with the fractions? I have read my text and our online lecture but obviously something isn't sinking in. The two equations that I am stuck on is as follows:

    1. -3x + 6y = 2

    2x + (2/3) y = 1

    2. (x/3) + (y/4) = 7
    (x/4) + (y/3) = 7

    Looking at #2, one can see that the answer is 12 but I am not clear on how it should be written.

    For # 1 what do I do with the 2/3?

    Thanks in advance for the help







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  2. #2
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    Hey!

     -3x +6y = 2 (a)
     2x + \frac{2}{3} y = 1 (b)

    Multiply second equation by -9.

     -3x + 6y = 2 (a)
     -18x - 6y = -9 (b)

    Add the two equations.

     (-3x + -18x) + (6y - 6y) = 2 - 9
     -21x = -7
     x = \frac{1}{3} .

    Can you do the rest?
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  3. #3
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    Are you trying to solve these equations simultaneously?

    -3x + 6y = 2
    \phantom{-}2x + \frac{2}{3}y = 1.

    Multiply the first equation by 2 and the second equation by 3

    -6x + 12y = 4
    \phantom{-}6x + 2y = 3

    Add the equations together

    (-6x + 12y) + (6x + 2y) = 4 + 3

    14y = 7

    y = \frac{1}{2}.

    Substitute this back into one of the original equations - the first will do.

    -3x + 6y = 2

    -3x + 6\left(\frac{1}{2}\right) = 2

    -3x + 3 = 2

    -3x = -1

    x = \frac{1}{3}.


    So the solution is (x, y) = \left(\frac{1}{3}, \frac{1}{2}\right).
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  4. #4
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    Hi, thank you so much, you have been a tremendous help to me. I think I can handle it from here :-)
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  5. #5
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    Wonderful, thank you very much. That is exactly what I needed to see, how it was broken down. How about the other one:

    (x/3) + (y/4) = 7
    (x/4) + (y/3) = 7

    I know the answer is 12 but I'm not sure how to write it out in the proper format.
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  6. #6
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    Sure thing! We're going to solve it basically the same way

     \frac{x}{3} + \frac{y}{4} = 7 (a)
     \frac{x}{4} + \frac{y}{3} = 7 (b)

    Multiply (a) by -3 and (b) by 4

     -x + \frac{-3y}{4} = -21 (a)
     x + \frac{4y}{3} = 28 (b)

    add a and b

     (-x + x) + \frac{-3y}{4} + \frac{4y}{3} = -21 + 28 = 7

    Now, solve for y and plug back in to either equation to solve back for x.
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  7. #7
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    You are the best, thank you again. After looking at it, it makes perfect sense, I think I tend to over analyze the problems, making them harder than they need to be. Once again, thank you and you have a great weekend.
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