Row reduction of augmented matrices and finding solutions

Question

Answer 1

a) The augmented matrix is

Second row of the matrices is zero.

The system is inconsistent. There is no solution.

C)

The first equation will be x  + 3z  = 18

Solving for x  = 18 - 3z

The second equation will be y  - z  = - 3

Solving for  y  = z  + 3

The other variables will be defined in terms of z  .

There fore z  will be the parameter and t  and the solution is 

x  = 18 - 3t, y  = t + 3 and z  = t.

Comments

Since 'a' has no solution does that mean x,y and z = 0?

---

No, it means that, it is impossible for the equation to be true no matter what value we assign to the variable, and graphs of the equations do not intersect .

.

Answer 2

d) The augmented matrix is

Continuing row reduction.

Since the second and third row are equal, the system has no solution and the system is inconsistent.

Answer 3

b) The augmented matrix is

The corresponding system of equation is x  + 2w  = 3

y  + w  = 4

z  - w  = 6

Since x , y  and  z  are leading variables and w  is free variable.

Solving leading variables in terms of free variable gives

x  = - 2w  + 3

y  = - w  + 4

z  = w  + 6

From this form of the equations we see the free variable w can be assigned an arbitary value say t, which then determines the values of leading variables x ,y  and z . Thus there are infinitely many solutions ,and the general solution is given by the

x  = - 2t  + 3

y  = - t  + 4

z  = t  + 6

w  = t .