If x + 2y = 20 and y + 2z = 9 and 2x + z = 22, what is the value of x + y + z

Thanks, Guest.......here's a slightly quicker method....

x + 2y = 20    →   y  = [20 - x ] /2     (1)

2x + z = 22   →   z  =  22 - 2x            (2)

y + 2z = 9    (3)

Substituting (1) and (2)  into (3), we have

[20 - x] / 2 + 2 [22 - 2x] = 9   multiply through by 2

20 - x  + 88 - 8x  = 18        simplify

-9x + 108  = 18      subtract 108 from each side

-9x = -90           divide both sides by -9

x = 10

And using (1) and (2)        y  = [20 - 10] / 2   = 5        and     z = 22 - 2(10)  =  2

Solve the following system:
{x+2 y = 20 |     (equation 1)
y+2 z = 9 |     (equation 2)
2 x+z = 22 |     (equation 3)
Swap equation 1 with equation 3:
{2 x+0 y+z = 22 |     (equation 1)
0 x+y+2 z = 9 |     (equation 2)
x+2 y+0 z = 20 |     (equation 3)
Subtract 1/2 × (equation 1) from equation 3:
{2 x+0 y+z = 22 |     (equation 1)
0 x+y+2 z = 9 |     (equation 2)
0 x+2 y-z/2 = 9 |     (equation 3)
Multiply equation 3 by 2:
{2 x+0 y+z = 22 |     (equation 1)
0 x+y+2 z = 9 |     (equation 2)
0 x+4 y-z = 18 |     (equation 3)
Swap equation 2 with equation 3:
{2 x+0 y+z = 22 |     (equation 1)
0 x+4 y-z = 18 |     (equation 2)
0 x+y+2 z = 9 |     (equation 3)
Subtract 1/4 × (equation 2) from equation 3:
{2 x+0 y+z = 22 |     (equation 1)
0 x+4 y-z = 18 |     (equation 2)
0 x+0 y+(9 z)/4 = 9/2 |     (equation 3)
Multiply equation 3 by 4/9:
{2 x+0 y+z = 22 |     (equation 1)
0 x+4 y-z = 18 |     (equation 2)
0 x+0 y+z = 2 |     (equation 3)
Add equation 3 to equation 2:
{2 x+0 y+z = 22 |     (equation 1)
0 x+4 y+0 z = 20 |     (equation 2)
0 x+0 y+z = 2 |     (equation 3)
Divide equation 2 by 4:
{2 x+0 y+z = 22 |     (equation 1)
0 x+y+0 z = 5 |     (equation 2)
0 x+0 y+z = 2 |     (equation 3)
Subtract equation 3 from equation 1:
{2 x+0 y+0 z = 20 |     (equation 1)
0 x+y+0 z = 5 |     (equation 2)
0 x+0 y+z = 2 |     (equation 3)
Divide equation 1 by 2:
{x+0 y+0 z = 10 |     (equation 1)
0 x+y+0 z = 5 |     (equation 2)
0 x+0 y+z = 2 |     (equation 3)
Collect results:
Answer: | 
| {x = 10
y = 5
z = 2

Just add up all three equations to get   3x + 3y + 3z = 51

Divide both sides by 3:

x + y + z = 17