Question

How do you solve for x in `(3/x) + (2/m) = (1/n)`?

Answer 1

`3/x=1/n-2/m=(m-2n)/(mn)`
`=>x/3=(mn)/(m-2n)`
`: . x =(3mn)/(m-2n)`

Answer 2

Manipulation shown in a lot of detail. Skip over the bits you know.

`x=(3mn)/(m-2n)`

Explanation:

Tricks for manipulating this equation type.

`color(red)("Very important "->" What you do to one side of the equation you do to the other side.")`

`color(green)("To move an add or subtract to the other side of the = turn it into 0")`
`color(green)("To move a multiply or divide to the other side of the = turn it into 1")`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given:`" "(3/x)+(2/m)=(1/n)`

Write as:
`" "3/x+2/m=1/n`

We need to determine the value of `x` so we need to 'get rid' of `2/m` so that we have `3/x` on its own.

Subtract `color(blue)(2/m) " "underline("from both sides")`

`" "color(brown)(3/x+2/m color(blue)(-2/m)" "=" "1/n color(blue)(-2/m))`
`color(white)(.)`

`" "3/x+0=1/n-2/m`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
To do the next bit we need to combine `1/n-2/m`

Multiply `1/n` by 1 but in the form of `1=m/m`
Multiply `2/m` by 1 but in the form of `1=n/n`

`" "3/x=(1/nxxm/m)-(2/mxxn/n)`

`" "3/x=((1xxm)/(nxxm))-((2xxn)/(mxxn))`

`" "3/x=((m)/(mn))-((2n)/(mn))`

`" "3/x=(m-2n)/(mn)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now you can turn everything upside down

`" "x/3=(mn)/(m-2n)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Multiply both sides by `color(blue)(" "3)`

`color(brown)(" "x xx(color(blue)(3))/3=(color(blue)(3)mn)/(m-2n))`

But `3/3 = 1` giving:

`" "x=(3mn)/(m-2n)`