The first number plus the third number is equal to the second number. the sum of the first number and the second number is six more than the third number. three times the first number minus two times the second number is equal to the third number. what is the sum of the three numbers?

I'll use x, y, and z for the numbers
x + z = y
x + y = 6 + z
3x - 2y = z

Substitute (x + z) for y in the second equation.
x + (x + z) = 6 + z and solve
2x = 6
x = 3

Rearrange another equation and substitute (3) for x.
(3) + y = 6 + z solve for y
y = 3 + z
substitute y in another equation
3(3) - 2(3+z) = z solve
9 - 6 - 2z = z
3 = -z
z = 3

going back to the first equation:
x+z=y
3 + 3 = 6
Answers:
x=3
y=6
z=3
check all equations to make sure it works for all equations