Fsc Part 1 Mathematics (Complete Solution)

If G is a group under the operation * and a,b∈ G, find the solutions of the equations:

a*x=b,                   x* a=b


Given that ''G'' is a group and a,b∈G  w.r.t* and a*x=b

As G is a group   => a-1=G


multiplying both sides by a-1

a-1 *(a*x)=a-1 * b

=> (a-1 * a)*x=a-1 *b                                    (Associative property)

e*x=a-1 * b

x=a-1 *b

a-1*b is solution of a * x=b

x * a=b

multiplying both sides by a-1

(x*a)*a-1 =a-1 *b

x*(a*a-1 )=a-1 *b

x*e=a-1 *b

x=a-1 *b

b*a-1is the solution of x * a=b

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