Three times the 1st number plus the 2nd number plus twice the 3rd is 5 is the same as 3x+y+2z=5. If three times the 2nd number is subtracted from the sum of the 1st and three times the 3rd number, the result is 2 is just x+3z-3y=2. And if the 3rd number is subtracted from two times the 1st number and three times the 2nd, giving a result of 1 means 2x+3y-z=1. Then you use substition on these equations to get a equation where one variable equals 2 others, like using the first to get y=5-2z-3x and then this can be substituted into the other two to get x+3z-3(5-2z-3x)=2 and 2x+3(5-2z-3x)-z=1 we can then simplify and subtract the equations. After simplification we have 10x+9z=17 and 7z+7x=16 which can be turned into 70x+63z=119 and 70x+70z=160 which can be then subtracted to get that 7z=41 and z=41/7. Now we backtrack to a two variable equation like 7z+7x=16 and plug in to find x. So after plugging in we get 41+7x=16 and 7x=-25 so x=-25/7. Now we choose a 3 variable equation and plug in. So taking y=5-2z-3x we plug in 41/7 for z and -25/7 for x to get y=5-82/7+75/7 and y=5-7/7 and y=4. Therefore x = -25/7 y = 4 and z = 41/7.
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