Question

Solve for R1. Thank you! :)

Answer 1

Answer:

$\displaystyle R_1 = \frac{R_T\cdot R_2}{R_2 - R_T}$

Step-by-step explanation:

We are given the equation:

$\displaystyle \frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}$

And we want to solve for R₁.

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

$\displaystyle R_TR_1R_2\left(\displaystyle \frac{1}{R_T}\right) = R_TR_1R_2\left(\frac{1}{R_1} + \frac{1}{R_2}\right)$

Multiply:

$\displaystyle R_1R_2 = R_TR_2 + R_TR_1$

Isolate R₁:

$\displaystyle R_1R_2 - R_1R_T = R_TR_2$

We can factor:

$\displaystyle R_1(R_2 - R_T) = R_TR_2$

And divide. Therefore, in conclusion:

$\displaystyle R_1 = \frac{R_T\cdot R_2}{R_2 - R_T}$

Answer 2

Answer:

$\displaystyle \rm R _{1} = \frac{ R _{T}R _{2} }{R _{2} - R _{T}}$

Step-by-step explanation:

Just an alternative.

we would like to solve the following equation for ${R_1}$.

$\displaystyle \rm \frac{1}{R _{T} } = \frac{1}{R _{1} } + \frac{1}{R _{2} }$

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

$\displaystyle \rm \frac{1}{R _{T} } = \frac{R _{2} + R _{1} }{R _{1} R _{2} }$

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:

$\displaystyle \rm R _{1} R _{2}= R _{T}(R _{2} + R _{1} )$

distribute:

$\displaystyle \rm R _{1} R _{2}= R _{T}R _{2} + R _{T} R _{1}$

isolate $R_1$ to the left hand side and change its sign:

$\displaystyle \rm R _{1} R _{2} - R _{T}R _{1}= R _{T}R _{2}$

factor out $R_1$ from the left hand side expression:

$\displaystyle \rm R _{1} (R _{2} - R _{T})= R _{T}R _{2}$

divide both sides by $R_2-R_T$:

$\displaystyle \rm \frac{R _{1} (R _{2} - R _{T})}{(R _{2} - R _{T})}= \frac{ R _{T}R _{2} }{(R _{2} - R _{T})}$

reduce fraction:

$\displaystyle \rm \boxed{ R _{1} = \frac{ R _{T}R _{2} }{R _{2} - R _{T}}}$

and we're done!