Systems of Equations in Three Variables?

Question

x - 2y + 2z = -1 x + 2y - z = 6 -3x + 6y - 6z = 3

Answer 1

The equations are 

x - 2y +2z = -1 - - - - > (1)

x + 2y - z  = 6  - - - - >(2)

-3x + 6y -6z =3- - - - >(3)

Solving this by gauss - Jordan method ,and writing the equations in matrix form,

Applying row operations 

This  system has infinite number of solutions.

Consider z= t,

The equations are x - 2y + 2z = -1 - - - - >(4)

                                  4y - 3z = 7 - - - - > (5)

                                        z = t. - - - - >(6)

Substitute (6) in (5) we get 

 4y - 3t = 7

4y = 7 +3t

y =(7+3t) / 4.             - - - -->(7)

Substituting (6) and (7) in (4)

x - 2y + 2z = -1

x -2(7 + 3t/4) +2t = -1

x - (7 + 3t/2) +2t = -1

2x - 7 - 3t +4t = -2

2x +t - 5 = 0

2x =(5- t)

x=(5-t)/2.

Hence the values of x  = (5-t)/2.

                            y =  (7+3t)/4

                            z = t .