Let the first number be x
Let the second number be y
Let the third number be z

the sum of 3 numbers is 5. 

X + y + z = 5

the first number minus the second plus the third is _____. 
(You left out this number, so you'll have to fill it in. 
I'll just leave a blank for now):  

x - y + z = ____

the first minus the third is 3 more than the second. 

x - z = y + 3

So you have this system of equations:

x + y + z = 5
x - y + z = ____
x - z = y + 3

Rearrange the third equation so like letters line
up vertically.

x + y + z = 5
x - y + z = ____
x - y - z = 3

I can't solve it because you left out the number in the
blank. 

If the number in the blank were 

-21, then the solution would be x = 4, y = 13, and z = -12.
-19, then the solution would be x = 4, y = 12, and z = -11.
-17, then the solution would be x = 4, y = 11, and z = -10.
-15, then the solution would be x = 4, y = 10, and z = -9.
-13, then the solution would be x = 4, y = 9, and z = -8.
-11, then the solution would be x = 4, y = 8, and z = -7.
-9, then the solution would be x = 4, y = 7, and z = -6.
-7, then the solution would be x = 4, y = 6, and z = -5.
-5, then the solution would be x = 4, y = 5, and z = -4.
-3, then the solution would be x = 4, y = 4, and z = -3.
-1, then the solution would be x = 4, y = 3, and z = -2.
1, then the solution would be x = 4, y = 2, and z = -1.
3, then the solution would be x = 4, y = 1, and z = 0.
5, then the solution would be x = 4, y = 0, and z = 1.
7, then the solution would be x = 4, y = -1, and z = 2.
9, then the solution would be x = 4, y = -2, and z = 3.
11, then the solution would be x = 4, y = -3, and z = 4.
13, then the solution would be x = 4, y = -4, and z = 5.
15, then the solution would be x = 4, y = -5, and z = 6.
17, then the solution would be x = 4, y = -6, and z = 7.
19, then the solution would be x = 4, y = -7, and z = 8.
21, then the solution would be x = 4, y = -8, and z = 9.

Edwin