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Math Help - linear equation with fraction

  1. #1
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    linear equation with fraction

    Hello, I am stuck on a couple of linear equations.
    Could someone please tell me the answer, then I will be able to see the method.

    Thanks kindly

    2/x + 1/2x = 7

    4/x+2 - 6/2x = 5/2x+4
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by fran1942 View Post
    Hello, I am stuck on a couple of linear equations.
    Could someone please tell me the answer, then I will be able to see the method.

    Thanks kindly

    2/x + 1/2x = 7

    4/x+2 - 6/2x = 5/2x+4
    As with all equations where there is a fraction you need to find a common denominator. For the first problem you have denominators x and 2x. What is the common denominator for these?

    -Dan
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  3. #3
    Junior Member lanierms's Avatar
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    1. \displaystyle\frac{2}{x}+\frac{1}{2x}=7

    \displaystyle\frac{4}{2x}+\frac{1}{2x}=7

    \displaystyle\frac{5}{2x}=7

    14x=5

    \displaystyle{x=\frac{5}{14}}.


    2. \displaystyle\frac{4}{x+2}-\frac{6}{2x}=\frac{5}{2x+4}

    \displaystyle\frac{8x-6(x+2)}{2x(x+2)}=\frac{5}{2x+4}

    \displaystyle\frac{2(x-6)}{2x(x+2)}=\frac{5}{2x+4}

    \displaystyle\frac{x-6}{x(x+2)}=\frac{5}{2x+4}

    5x(x+2)=2(x-6)(x+2)

    since, x+2\not=0

    5x=2(x-6)

    \therefore x=-4.
    Last edited by lanierms; February 28th 2011 at 04:02 AM.
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  4. #4
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    Quote Originally Posted by lanierms View Post
    ...

    2. \displaystyle\frac{4}{x+2}-\frac{6}{2x}=\frac{5}{2x+4}

    \displaystyle\frac{8x-6(x+2)}{2x(x+2)}=\frac{5}{2x+4}

    \displaystyle\frac{2(x+1)}{2x(x+2)}=\frac{5}{2x+4}

    ...
    I don't want to pick at you but the last quoted line is definitely wrong:

    \displaystyle\frac{8x-6(x+2)}{2x(x+2)}=\frac{5}{2x+4}~\implies~\displays  tyle\frac{8x-6x-12}{2x(x+2)}=\frac{5}{2x+4}

    By the way: You should have allowed the OP to do a little bit of the question by himself.
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  5. #5
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    Quote Originally Posted by lanierms View Post
    5x(x+2)=2(x+1)(x+2)
    EASIER this way:
    at this point, divide both sides by (x + 2):
    5x = 2x + 2
    3x = 2
    x = 2/3
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  6. #6
    Junior Member lanierms's Avatar
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    Quote Originally Posted by earboth View Post
    I don't want to pick at you but the last quoted line is definitely wrong:

    \displaystyle\frac{8x-6(x+2)}{2x(x+2)}=\frac{5}{2x+4}~\implies~\displays  tyle\frac{8x-6x-12}{2x(x+2)}=\frac{5}{2x+4}

    By the way: You should have allowed the OP to do a little bit of the question by himself.
    Whoops. Sorry, made a mistake. ............. I edited my solution.
    Last edited by lanierms; February 28th 2011 at 04:03 AM.
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  7. #7
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    Quote Originally Posted by fran1942 View Post
    Hello, I am stuck on a couple of linear equations.
    Could someone please tell me the answer, then I will be able to see the method.

    Thanks kindly

    2/x + 1/2x = 7

    4/x+2 - 6/2x = 5/2x+4
    the best hint for this problem would be to assume 1/x as X and 1/2x as Y. so therefore the equation now becomes as 2X + Y = 7
    and 4X + 2 - 6Y = 5Y + 4

    the second equation can be further simplified as 4X - 11Y = 2 .
    let the first equation 2X + Y = 7 be (1)
    and second equation 4X - 11Y = 2 be (2)
    now (1) x 4 and (2) x 2 , and the equation becomes
    8X + 4Y = 28
    8X + 22Y = 4
    subtracting these two equations we get -18Y = 24
    therefore Y is - 24/18 = - 4/3
    applying Y = - 4/3 in (1) we get 8X - 16/3 = 28
    8X = 28 + 16/3
    = (84 + 16)/3 = 100/3
    X= 100/ (3x8)
    = 25/12
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  8. #8
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    Quote Originally Posted by krishnan View Post
    the best hint for this problem would be to assume 1/x as X and 1/2x as Y. so therefore the equation now becomes as 2X + Y = 7
    and 4X + 2 - 6Y = 5Y + 4
    the second equation can be further simplified as 4X - 11Y = 2 .
    let the first equation 2X + Y = 7 be (1)
    and second equation 4X - 11Y = 2 be (2)
    No, not a "system", but 2 different problems.
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  9. #9
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    Quote Originally Posted by Wilmer View Post
    No, not a "system", but 2 different problems.
    oh! then it is much simpler. the problem is 2/x + 1/2x = 7 . take 1/x as X , so the equation now becomes as 2X + X/2 = 7
    now make the denominator common on the left hand side of the equation ie. (4X + X)/2 = 7
    now this becomes (4X + X) = 14 . so therefore 5X = 14 , X = 14/5
    as a result X = 14/5 = 1/x , x= 5/14
    the same procedure can be applied for the second problem too.
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  10. #10
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    Quote Originally Posted by Wilmer View Post
    EASIER this way:
    at this point, divide both sides by (x + 2):
    5x = 2x + 2
    3x = 2
    x = 2/3
    Provided, of course, that x+ 2 is not 0. Which means there is another solution.
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