Question

Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination.If the system has an...

Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination.If the system has an infinite number of solutions, express x, y, and z in terms of the parameter t.)

3x + 3y + 9z = 6

x + y + 3z = 2

2x + 5y + 15z = 10

-x + 2y + 6z = 4

(x, y, z) = ?

Augmented matrix for given system of equations

X1 X2 X3 b
1 3 3 9 6
2 1 1 3 2
3 2 5 15 10
4 -1 2 6 4

Find the pivot in the 1st column and swap the 2nd and the 1st rows

X1 X2 X3 b
1 1 1 3 2
2 3 3 9 6
3 2 5 15 10
4 -1 2 6 4

Eliminate the 1st column

X1 X2 X3 b
1 1 1 3 2
2 0 0 0 0
3 0 3 9 6
4 0 3 9 6

Make the pivot in the 2nd column by dividing the 3rd row by 3 and swap the 3rd and the 2nd rows

X1 X2 X3 b
1 1 1 3 2
2 0 1 3 2
3 0 0 0 0
4 0 3 9 6

Eliminate the 2nd column

X1 X2 X3 b
1 1 0 0 0
2 0 1 3 2
3 0 0 0 0
4 0 0 0 0

Solution set:

x = 0

y = 2 - 3t

z = t, t = free parameter

(x,y,z) = (0, 2 - 3t, t) (answer)