Math problem

We have 6 equations: 

$a+b=10$    (1)

$a+c=18$    (2)

$a+d=19$    (3)

$b+c=22$     (4)

$b+d=23$    (5)

$c+d=31$    (6)

Using equations 2 and 3, we see that $d = 1+c$

Subbing this into (6), we get: $c+c+1=31$, meaning $c=15$

Subbing this into (2), we find $a=3$

Subbing the value of c into (4), we find $b=7$

Subbing the value of c into (6), we find $d = 16$

Thus, the values are $\color{brown}\boxed{3,7,15,16}$

Let    a,b, c, d   be in increasing order

Using some logic....we know that 

a + b  = 10       (1)

a + c =  18       (2)

c + d =  31      (3)

b + d =  23      (4)

What we don't know is  that if

a + d =  19   or   a + d  =  22  or   b + c = 19  or   b + c =   22

To see which is true    subtract   (4)  from (1)    and we  get 

a - d  =  -13        (5)

Now let us assume that   a + d  =  22      (6)

Add (5) and (6)  and we get that

2a = 9             but a here is not an integer

So....it must be that

a + d = 19

So

a - d = -13

a + d = 19          add these

2a  = 6

So.....using our equations.....

a = 3

b = 7

c = 15

d = 16