Anumber is equal to twice a smaller number plus 3. the same number is equal to twice the sum of the smaller number and 1. how many solutions are possible for this situation?

If a number (lets say 'y' is that number) is equal to (I.e. y = )..mathematically
now y is equal to twice the smaller number ( lets say that the smaller number is 'x' ) plus (+) 3 . that brings us to the linear equation...
y = 2x + 3;

also the second part of the question says that the same number ( I.e. 'y' here ) is twice the sum of smaller number (I.e. '2x' here ) and 1.

which gives us 2nd equation I.e.
y = 2x + 1 ;

the question is to find the possible number of solutions for both these equation

to find solution means to find all those points or say numbers which would satisfy both these equations

now for the first equation..
lets say ( y = 5 ) this means ( 2(1) + 3 )
so the solution will be ( 1, 5)
now for the second equation..
we will proceed with whatever data we know..
I.e. we know that value of ' y ' is 5, as we assumed

putting value of y in 2 nd equation we get ( 2(2)+1 )
I.e solution will be ( 2, 5)

now what does this signify...?
it signifies that there is no such point that will satisfy both these equations simultaneously..

hence we can say that there is no sokution possilble to this situation.

It’s hard to understand what you put.. can you show a picture ?