The sum of three numbers is 6. the third number is the sum of the first and second numbers. the first number is one more than the third number. find the numbers

x is the first number
y is the second number
z is the third number 
Given
x + y + z = 6 (The sum of three numbers is 6)
z = x + y (the third number is the sum of the first and second numbers)
x = z + 1  (the first number is one more than the third number)

x + y + z=6
z + z  = 6 (because z = x + y)
2 z = 6

Divide both sides by 2 then
z = 3

Replace z = 2 into x = z+1 to find x
x = 3 + 1
x = 4

Replace x = 4 , z = 3 into z = x + y to find y 
3 = 4 + y

Subtract 4 to both sides
3 - 4 = 4 + y -4

Simplify
-1 = y or y = -1

Answer x = 4, y = -1, z = 3 

Proof:
     x + y + z = 6 (The sum of three numbers is 6)
4 + (-1) + 3  = 6
                6  = 6

 z = x + y (the third number is the sum of the first and second numbers)
 3 = 4 + (-1)
 3 = 3

 x = z + 1  (the first number is one more than the third number)
 4 = 3 + 1
4  = 4