Question

Given `(x/a) + (y/b) + (z/c) = 1`, how do you solve for a?

Answer 1

`color(blue)(a=(-bcx)/(cy+bz-bc)`

Explanation:

`(x/a)+(y/b)+z/c=1`

multiply L.H.S. and R.H.S. by abc

`:.(x(cancela^color(blue)1bc))/cancela^color(blue)1+(y(acancelb^color(blue)1c))/cancelb^color(blue)1+(z(abcancelc^color(blue)1))/cancelc^color(blue)1=abc`

`:.bcx+acy+abz=abc`

`:.acy+abz-abc=-bcx`

`:.a(cy+bz-bc)=-bcx`

`:.color(blue)(a=(-bcx)/(cy+bz-bc)`

Answer 2

`a=(-bcx)/(cy+bz-bc)`

Explanation:

A good strategy is to get rid of the fractions first. You can easily do this by multiplying both sides by `a*b*c`.

`abc (x/a)+abc (y/b)+abc (z/c)=abc (1)`

`bcx+acy+abz=abc`

Next, collect all the terms with `a` together on one side.

`acy+abz-abc=-bcx`

Factor out the common `a` on the left.

`a(cy+bz-bc)=-bcx`

Divide both sides by the expression without `a` on the left.

`a=(-bcx)/(cy+bz-bc)`