Question

Let a, b, c and x elements in the group G. In each of the following solve for x in terms of a, b, and c.

Solve axb =c

Answer 1

`x=c.a^(-1).b^(-1)`

Explanation:

Given equation,

`axb = c-----(1)`

`(axb) ( a^(-1).b^(-1) ) = c.(a^(-1)b{-1})`

`(xb)(a)(a^(-1).b^(-1) ) = c.(a^(-1)b{-1})` ( commutative law of multiplication )

`(xb)(a.a^(-1)).b^(-1) = c.a^(-1)b{-1}` ( Associative property )

`(xb).e.b^(-1) = c.a^(-1)b{-1}`

`(xb).b^(-1)=c.a^(-1).b^(-1)` ( identity property )

`x.(b.b^(-1))=c.a^(-1).b^(-1)` ( Associative property )

`x.e=c.a^(-1).b^(-1)`

`x=c.a^(-1).b^(-1)` ( identity property )