Question

The binary operation *on the set R of all real numbers is defined as a*b=2a+3b-5.
a)find the inverse element of*
b)show whether or not *is commutative
c)find -3 *1/4

Answer 1

Is * commutative?
a*b = 2a+3b-5
b*a = 2b+3a-5
I think you can see that in general, * is not commutative.

In order to have an inverse, you need an identity c, such that
a*c = a
For example,
the additive identity is 0, so the additive inverse of a is -a
the multiplicative identity is 1, so the mult. inverse of a is 1/a
so, we need
2a+3c-5 = a
c = (5-a)/3
So, use that to find the inverse of a.