Let

x = the first number

y = the second number

z = the third number

The sum of three numbers is 9:

x +y + z = 9

The first number minus the second plus the third is 3:

x - y + z = 3

The first minus the third is 1 more than the second:

x - z = y + 1

We have the following system of equations:

x +y + z = 9

x - y + z = 3

x - z = y + 1

........

........

x = 5

y = 3

z = 1

The numbers are 5, 3 and 1.