Question

Solve for R1. Thank you! :)

Answer 1

`\displaystyle \rm R _(1) = ( R _(T)R _(2) )/(R _(2) - R _(T))`

Explanation:

Just an alternative.

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

`\displaystyle \rm (1)/(R _(T) ) = (1)/(R _(1) ) + (1)/(R _(2) )`

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

`\displaystyle \rm (1)/(R _(T) ) = (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 (R _(1) (R _(2) - R _(T)))/((R _(2) - R _(T)))= ( R _(T)R _(2) )/((R _(2) - R _(T)))`

reduce fraction:

`\displaystyle \rm \boxed{ R _(1) = ( R _(T)R _(2) )/(R _(2) - R _(T))}`

and we're done!

Answer 2

`\displaystyle R_1 = (R_T\cdot R_2)/(R_2 - R_T)`

Explanation:

We are given the equation:

`\displaystyle (1)/(R_T) = (1)/(R_1) + (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 (1)/(R_T)\right) = R_TR_1R_2\left((1)/(R_1) + (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 = (R_T\cdot R_2)/(R_2 - R_T)`