⎛ -1 2 3 ⎞
⎜ 2 -3 5 ⎜represents the LHS of the equations.
⎝ 2 1 3 ⎠
The determinant of this is -1(-9-5)-2(6-10)+3(2+6)=14+8+24=46. We need this to find x, y and z.
Now the "x" determinant:
⎜ 13 2 3 ⎜
⎜ 10 -3 5 ⎜= 13(-14)-2(30-60)+3(10+36) = -182+60+138 = 16. x=16/46=8/23.
⎜ 12 1 3 ⎜
The "y" determinant:
⎜ -1 13 3 ⎜
⎜ 2 10 5 ⎜= -1(-30)-13(-4)+3(24-20) = 30+52+12 = 94. y=94/46=47/23.
⎜ 2 12 3 ⎜
The "z" determinant:
⎜ -1 2 13 ⎜
⎜ 2 -3 10 ⎜= -1(-36-10)-2(4)+13(8) = 46-8+104 = 142. z=142/46=71/23.
⎜ 2 1 12 ⎜
Answer: (x,y,z)=(8/23,47/23,71/23).