There are many ways to do this, but here is one of them.
Convert
4a+5b-6c=2
-3a-2b+7c=-15
-a+4b+2c=-13
to matrix form to get
4 5 -6 2
-3 -2 7 -15
-1 4 2 -13
Then use row reduction
4 5 -6 2
-3 -2 7 -15
-1 4 2 -13
1 1.25 -1.5 0.5 0.25*R1
-3 -2 7 -15
-1 4 2 -13
1 1.25 -1.5 0.5
0 1.75 2.5 -13.5 R2 + (3)*R1
-1 4 2 -13
1 1.25 -1.5 0.5
0 1.75 2.5 -13.5
0 5.25 0.5 -12.5 R3 + (1)*R1
1 1.25 -1.5 0.5
0 1 1.4286 -7.7143 0.571429*R2
0 5.25 0.5 -12.5
1 0 -3.2857 10.1429 R1 + (-1.25)*R2
0 1 1.4286 -7.7143
0 5.25 0.5 -12.5
1 0 -3.2857 10.1429
0 1 1.4286 -7.7143
0 0 -7 28 R3 + (-5.25)*R2
1 0 -3.2857 10.1429
0 1 1.4286 -7.7143
0 0 1 -4 -0.142857*R3
1 0 0 -3 R1 + (3.28571)*R3
0 1 1.4286 -7.7143
0 0 1 -4
1 0 0 -3
0 1 0 -2 R2 + (-1.42857)*R3
0 0 1 -4
The right hand column is -3, -2, -4
These 3 values correspond to a, b, and c respectively.
So the solutions are...
a = -3
b = -2
c = -4
They form the ordered triple (-3, -2, -4)