Find the ordered triple of these equations. x - 2y + 3z = -3 2x - 3y - z = 7 3x + y - 2z = 6?

x - 2y + 3z = -3 …… [1]
2x - 3y - z = 7 …… [2]
3x + y - 2z = 6 …… [3]

[1]*(-2) :
-2x + 4y - 6z = 6 …… [4]

[1]*(-3) :
-3x + 6y - 9z = 9 …… [5]

[2] + [4] :
y - 7z = 13
y = 7z + 13 …… [6]

[3] + [5] :
7y - 11z = 15 …… [7]

Substitute [6] into [7] :
7*(7z + 13) - 11z = 15
49z + 91 - 11z = 15
38z = -76
z = -2

Substitute z = -2 into [6] :
y = 7(-2) + 13
y = -14 + 13
y = -1

Substitute y = -1 and z = -2 into [1] :
x - 2(-1) + 3(-2) = -3
x + 2 - 6 = -3
x - 4 = -3
x = 1

Hence, (x, y, z) = (1, -1, -2)

{x, y, z} = {1, -1, -2}

x - 2y + 3z = -3 and 2x - 3y - z = 7 and 3x + y - 2z = 6, so
x - 2y + 3z = -3 and y - 7z = 13 and 7y - 11z = 15, so
y = (13×-11+7×15)/(-11+7×7) = -1 and z = (15-13×7)/(-11+7×7) = -2 and so x - 2×-1 + 3×-2 = -3 which yields x = 1

The ordered triple of these equations (the common point of the three given planes) is (1, -1, -2).