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Math Help - Combining two equations

  1. #1
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    Combining two equations

    Hi,

    I have two equations as follows:

    L is proportional to (M^6/R^7)

    which I have to combine with:

    L is proportional to M^5.5 R^-0.5

    I need to end up with a relationship which has L proportional to R so that M is eliminated.

    My current attempt has gone along the following lines:

    Rearrange 2nd Equation so that M is subject and then substitute L back into the equation to have:

    M is proportional to ((M^6/R^7)/R^-0.5) <<<<this would be taking to the root 5.5 as well but wasn't sure how to write this down to make it clear on the screen.

    My main problem is eliminating M from the equation. Any help appreciated!
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  2. #2
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    Crna Gora
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    Re: Combining two equations

    L=k_1 \cdot \frac{M^6}{R^7} ~\text {and}~L=k_2 \cdot \frac{M^{5.5}}{R^{0.5}}

    k_1 \cdot \frac{M^6}{R^7}=k_2 \cdot \frac{M^{5.5}}{R^{0.5}} \Rightarrow k_1 \cdot M^{0.5}=k_2 \cdot R^{6.5} \Rightarrow M=\left(\frac{k_2}{k_1}\right)^2 \cdot R^{13}

    L=k_1 \cdot \left(\frac{k_2}{k_1}\right)^{12} \cdot \frac{R^{78}}{R^7} \Rightarrow L=k_3 \cdot R^{71}
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  3. #3
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    Re: Combining two equations

    Hello, ohdearthisisheadrot!

    I have two equations as follows:
    . . L is proportional to (M^6/R^7)
    . . L is proportional to M^5.5 R^-0.5

    I need to end up with a relationship which has L proportional to R so that M is eliminated.

    To eliminate M, solve both equations for M and equate.

    We have: . \begin{Bmatrix}L &=& a\dfrac{M^6}{R^7} \\ \\[-3mm] L &=& b\dfrac{M^{\frac{11}{2}}}{R^{\frac{1}{2}}} \end{Bmatrix} \quad\Rightarrow\quad \begin{Bmatrix}M^6 &=& \dfrac{LR^7}{a} & [1] \\ \\[-3mm] M^{\frac{11}{2}} &=& \dfrac{LR^{\frac{1}{2}}}{b} & [2] \end{Bmatrix}


    \begin{array}{cccccccccccc}\text{Raise [1] to the power }\frac{11}{2}\!: & \left(M^6\right)^{\frac{11}{2}} &=& \left(\dfrac{LR^7}{a}\right)^{\frac{11}{2}} & \Rightarrow & M^{33} &=& \dfrac{L^{\frac{11}{2}}R^{\frac{77}{2}}}{a^{\frac{  11}{2}}} & [3]\\ \\  \text{Raise [2] to the power }6\!: & \left(M^{\frac{11}{2}}\right)^6 &=& \left(\dfrac{LR^{\frac{1}{2}}}{b}\right)^6  & \Rightarrow & M^{33} &=& \dfrac{L^6R^3}{b^6}& [4] \end{array}


    Equate [3] and [4]: . \dfrac{L^{\frac{11}{2}}R^{\frac{77}{2}}}{a^{\frac{  11}{2}}} \;=\;\dfrac{L^6R^3}{b^6} \quad\Rightarrow\quad b^6L^{\frac{11}{2}}R^{\frac{77}{2}} \;=\;a^{\frac{11}{2}} L^6 R^3

    Divide by L^{\frac{11}{2}}R^3\!:\;\;b^6R^{\frac{71}{2}} \;=\;a^{\frac{11}{2}} L^{\frac{1}{2}} \quad\Rightarrow\quad L^{\frac{1}{2}} \;=\;\dfrac{b^6R^{\frac{71}{2}}}{a^{\frac{11}{2}}}

    Square both sides: . L \;=\;\frac{b^{12}}{a^{11}}R^{71}
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  4. #4
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    Re: Combining two equations

    Sorry my mistake on the initial question:

    L = M^5.5*R^-0.5 not L = M^5.5/R^-0.5

    does this make a difference?

    thanks for your help!
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