Answer :

xKelvin

Answer:

[tex]\displaystyle R_1 = \frac{R_T\cdot R_2}{R_2 - R_T}[/tex]

Step-by-step explanation:

We are given the equation:

[tex]\displaystyle \frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}[/tex]

And we want to solve for R₁.

We can multiply everything by the three denominators to remove all fractions:

[tex]\displaystyle R_TR_1R_2\left(\displaystyle \frac{1}{R_T}\right) = R_TR_1R_2\left(\frac{1}{R_1} + \frac{1}{R_2}\right)[/tex]

Multiply:

[tex]\displaystyle R_1R_2 = R_TR_2 + R_TR_1[/tex]

Isolate R₁:

[tex]\displaystyle R_1R_2 - R_1R_T = R_TR_2[/tex]

We can factor:

[tex]\displaystyle R_1(R_2 - R_T) = R_TR_2[/tex]

And divide. Therefore, in conclusion:

[tex]\displaystyle R_1 = \frac{R_T\cdot R_2}{R_2 - R_T}[/tex]

Nayefx

Answer:

[tex]\displaystyle \rm R _{1} = \frac{ R _{T}R _{2} }{R _{2} - R _{T}}[/tex]

Step-by-step explanation:

Just an alternative.

we would like to solve the following equation for [tex]{R_1}[/tex].

[tex] \displaystyle \rm \frac{1}{R _{T} } = \frac{1}{R _{1} } + \frac{1}{R _{2} } [/tex]

in order to do so,we can simplify the right hand side which yields:

[tex] \displaystyle \rm \frac{1}{R _{T} } = \frac{R _{2} + R _{1} }{R _{1} R _{2} } [/tex]

Steps, used to simplify the right hand side:

  1. find the LCM of the denominators of the fractions i.e LCM(R_1,R_2)=R_1•R_2
  2. divide the LCM by the denominator of every fraction
  3. multiply the result of the division by the numerator of every fraction

Cross multiplication:

[tex]\displaystyle \rm R _{1} R _{2}= R _{T}(R _{2} + R _{1} )[/tex]

distribute:

[tex]\displaystyle \rm R _{1} R _{2}= R _{T}R _{2} + R _{T} R _{1} [/tex]

isolate [tex]R_1[/tex] to the left hand side and change its sign:

[tex]\displaystyle \rm R _{1} R _{2} - R _{T}R _{1}= R _{T}R _{2} [/tex]

factor out [tex]R_1[/tex] from the left hand side expression:

[tex]\displaystyle \rm R _{1} (R _{2} - R _{T})= R _{T}R _{2} [/tex]

divide both sides by [tex]R_2-R_T[/tex]:

[tex]\displaystyle \rm \frac{R _{1} (R _{2} - R _{T})}{(R _{2} - R _{T})}= \frac{ R _{T}R _{2} }{(R _{2} - R _{T})}[/tex]

reduce fraction:

[tex]\displaystyle \rm \boxed{ R _{1} = \frac{ R _{T}R _{2} }{R _{2} - R _{T}}}[/tex]

and we're done!

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