If there is a unique left identity, then it is also a right identity
Solution 1:
Let $b \in R$. Then
$$(be-b+e)a=a \forall a \in R$$
By the uniqueness you get $$be-b+e=e$$
As $b \in R$ is arbitrary, you are done.
Let $b \in R$. Then
$$(be-b+e)a=a \forall a \in R$$
By the uniqueness you get $$be-b+e=e$$
As $b \in R$ is arbitrary, you are done.