Algebraic Manipulation

Hi,

I am struggling to rearrange the equation for

A_1=\dfrac{B_1}{1-B_2}

into

B_1=\dfrac{A_1}{1+A_2}

. I also have the information that

A_2=\dfrac{B_2}{1-B_2}

which I know how to translate into

B_2=\dfrac{A_2}{1+A_2}

. Could someone please show me how to do this. Thanks.

(edited 13 years ago)

Well, take

B_2=\dfrac{A_2}{1+A_2}

and substitute into

A_1=\dfrac{B_1}{1-B_2}

to give

A_1=\dfrac{B_1}{1-\dfrac{A_2}{1+A_2}}

.

Now multiply by the denominator and rearrange:

B_1=A_1-\dfrac{A_1A_2}{1+A_2}=\dfrac{A_1(1+A_2)}{1+A_2}-\dfrac{A_1A_2}{1+A_2}=\dfrac{A_1+A_1A_2-A_1A_2}{1+A_2}=\dfrac{A_1}{1+A_2}

Well, take

B_2=\dfrac{A_2}{1+A_2}

and substitute into

A_1=\dfrac{B_1}{1-B_2}

to give

A_1=\dfrac{B_1}{1-\dfrac{A_2}{1+A_2}}

.

Now multiply by the denominator and rearrange:

B_1=A_1-\dfrac{A_1A_2}{1+A_2}=\dfrac{A_1(1+A_2)}{1+A_2}-\dfrac{A_1A_2}{1+A_2}=\dfrac{A_1+A_1A_2-A_1A_2}{1+A_2}=\dfrac{A_1}{1+A_2}

This is the gift that keeps on giving. Thanks for your input, Sonny Jim. You've just saved my bacon. <3

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