Make determinant Δ from the coefficients of I1, I2, I3:
| 1 1 1 |
| 8 0 -10⎜= 1(0-30)-1(0+60)+1(-24-0)=-30-60-24=-114
| 6 -3 0 |
3.1) Now make the determinant for I2, using the column of constants in place of the variable's column:
Δ2=
| 1 0 1 |
| 8 8 -10 | = 1(0+120)+0+1(96-48)=120+48=168
| 6 12 0 |
I2=Δ2/Δ=168/-114=-28/19.
3.2) 8I1-10I3=8, 6I1-3I2=12⇒①4I1-5I3=4, ②2I1-I2=4; ③I1+I2+I3=0;
②+③=3I1+I3=4⇒④15I1+5I3=20;
①+④=19I1=24, I1=24/19⇒I3=4-3I1=4-72/19=4/19.
(We can deduce that I2=-(I1+I3)=-28/19 which confirms 3.1) I2=-28/19.)