\(\text{If $3a + b + c = -3, a+3b+c = 9, a+b+3c = 19$, then find $abc$. }\)
+129852 3a + b + c = -3 (1)
a + 3b + c = 9 (2)
a + b + 3c = 19 (3)
Multiply (1) through by -3 and we get -9a - 3b - 3c = 9
Add this to (2) and we get that -8a - 2c = 18 (4)
Multiply (3) through by -3 and we get that -3a - 3b -9c = -57
Add this to (2) and we get that -2a - 8c = -48 (5)
Add (4) and (5) and we get that
-10a - 10c = -30 divide through by -10
a + c = 3
Sub this into (2) to find b
3b + 3 = 9
3b = 6
b = 2
And since a + c = 3 then c = 3 - a
Sub this into (3) to find a
a + 2 + 3(3 - a) = 19
a + 2 + 9 - 3a = 19
-2a + 11 = 19
-2a = 8
a = - 4
And using (1)
3(-4) + 2 + c = -3
-10 + c = -3
c = 7
So {a, b, c} = { -4, 2, 7 }
So abc = (-4)(2)(7) = -56
