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Thread: Solve an equation system

  1. #1
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    Solve an equation system

    Please solve this equation system:
    x^2+xy=1/9
    y^2+xy=1/16
    Thank you!
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  2. #2
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    What ideas have you had so far?
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  3. #3
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    First i tried to square everything, however could not continue to solve it after that.
    Later on I tried to factorize the first two terms in both equations, but I could not do this. My third attempt was to divide every term my x and use the substitution method. The last try was to use the substitution method directly.
    All my attempts gave an incorrect answer or I was not able to continue.
    Can you please help me?
    Thank you!
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  4. #4
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    Rearrange equation 1:

    $\displaystyle \displaystyle xy = \frac{1}{9} - x^2$

    $\displaystyle \displaystyle xy = \frac{1 - 9x^2}{9}$

    $\displaystyle \displaystyle y = \frac{1 - 9x^2}{9x}$.


    Now substitute this into the second equation and solve for $\displaystyle \displaystyle x$...
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  5. #5
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    Thank you prove it!
    I substituted the value into the equation, however I did not get the correct answer. I got x=4/21, but the answer should be 4/15 or -4/15. Can you please show me how to solve it?
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  6. #6
    Junior Member lanierms's Avatar
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    $\displaystyle x^2+xy=\frac{1}{9}$ - - - - - - - - - - 1.

    $\displaystyle y^2+xy=\frac{1}{16}$ - - - - - - - - - - 2.

    From 1., we get $\displaystyle x(x+y)=\frac{1}{9}$

    Therefore, $\displaystyle x+y=\frac{1}{9x}$

    From 2., we get $\displaystyle y(x+y)=\frac{1}{16}$

    Therefore, $\displaystyle x+y=\frac{1}{16y}$

    So, $\displaystyle \frac{1}{9x}=\frac{1}{16y}$

    $\displaystyle x:y=16:9$

    Now, let's say that $\displaystyle x=16k$, $\displaystyle y=9k$.

    From 1,. $\displaystyle 16k(16k+9k)=\frac{1}{9}$

    $\displaystyle k^2=\frac{1}{9\times16\times25}$

    $\displaystyle k=\pm\frac{1}{60}$

    So, $\displaystyle x=\pm\frac{4}{15}$, $\displaystyle y=\pm\frac{3}{20}$.
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  7. #7
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    Thank you very much lanierms!
    I understand how to solve it now.
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  8. #8
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    I hope you do. However, you should understand that you do not yet have the "solution". The problem does not ask for separate values of x and y but for pairs that satisfy the equation.
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  9. #9
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    I do not understand HallsofIvy. Can you please explain. When I look in the key, it shows x and y values as the answers.
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