Question

If a(x+b)=bx-c, what is x?

Answer 1

expand the bracket

`ax+ab=bx-c`

make `x` the subject

`ax+ab+c=bx`

`ab+c=bx-ax`

`ab+c=x(b-a)`

`(ab+c)/(b-a)=x`

Answer 2

`x = (c + ab)/(b - a)`

Explanation:

We have,

`color(white)(xxx)a(x + b) = bx - c`

`rArr ax + ab = bx -c` [Distributive Property]

`rArr ax +ab - bx = cancel(bx) cancel(- bx) - c` [Subtract `bx` from both sides]

`rArr ax - bx + cancel(ab) cancel(-ab) = -c - ab`

`rArr x(a- b) = (-c - ab)` [Group like terms]

`rArr x = (c + ab)/-(a - b)` [Divide both sides by `(a - b)`] [Only if `a != b`]

`rArr x = (c + ab)/(b - a)`

If `a = b`, and `ab + c != 0`, Then, The solution doesn't exist.

If `c lt 0`, `a = b` and `ab + c = 0`, Then `x in RR`

And If, `c gt 0`, `a = b` and `ab + c = 0`, Then Again, Solution is a Complex Number, so No Real Solutions.

And Gotcha.

Hope this helps.

Answer 3

If `a=b` and `ab+c=0` then we get all real numbers as solution.

if `a=b` and `ab+cne0` then no solution
if `ane b` then `x=(ab+c)/(b-a)`

Explanation:

Write your equation in the form

`x(b-a)=ab+c`
and distinguish the cases above.