a + b + 0c = 1
0a + 2b + c = 1
a + 0b + 2c = 1
In the first equation, subtract out the third equation:
(also, don't write the third equation again, since we've used it)
0a + b - 2c = 0
0a + 2b + c = 1
Multiply the second equation by 2 on both sides:
0a + b - 2c = 0
0a + 4b + 2c = 2
Add the first and second equation together:
0a + 5b + 0c = 2
b = 2/5
But since we know that b - 2c = 0 (see 5 lines above), we can do this:
2/5 - 2c = 0
-2c = -2/5
c = 1/5
Since we know a + b = 1 (original problem), we can do this:
a + 2/5 = 1
a = 3/5
Answer: a = 3/5, b = 2/5, c = 1/5