Question 117615
1.{{{A + B + C = 0}}}
2.{{{4A - 10B = 3}}}
3.{{{-10B + 5C = 6}}}
The quickest and easiest way is to use equations 2 and 3 and find expressions for A and C in terms of B and then substitute those into equation 1. 
You then solve for B and go back and solve for A and C.
2.{{{4A - 10B = 3}}}
{{{4A=3+10B}}}
{{{A=(3+10B)/4}}}
3.{{{-10B + 5C = 6}}}
{{{5C=6+10B}}}
{{{C=(6+10B)/5}}}
Now go back to equation 1.
1.{{{A + B + C = 0}}}
{{{(3+10B)/4 + B +(6+10B)/5 = 0}}}
Let's multiply both sides by (4*5) to get rid of the denominators.
{{{(3+10B)/4 + B +(6+10B)/5 = 0}}}
{{{5(3+10B)+20B+4(6+10B)=0}}}
{{{15+50B+20B+24+40B=0}}}
{{{110B+39=0}}}
{{{110B=-39}}}
{{{highlight(B=-39/110)}}}
From our derivation above for A,
{{{A=(3+10B)/4}}}
{{{4A=3+10(-39/110)}}}
{{{4A=330/110-390/110)}}}
{{{4A=-60/110)}}}
{{{A=-15/110)}}}
{{{highlight(A=-3/22))}}}
From equation 1, 
{{{A+B+C=0}}}
{{{-15/110-39/110+C=0}}}
{{{C=(39+15)/110}}}
{{{C=54/110}}}
{{{highlight(C=27/55)}}}