# Problem4:

Find the solution to the system of equations with augmented matrix given by,

> ;

```                           [1    0    0    -7     8]
[                       ]
[0    1    0     3     2]
[                       ]
[0    0    1     1    -5]```

### SOLUTION:

 Assume the unknowns are (w,x,y,z). Start from the last row working upwards. The last equation is,

> Eq3 := y+z = -5;

`                               Eq3 := y + z = -5`
 from where we get that

> y := -5 -z:
 The second equation is,

> Eq2 := x + 3* z = 2;

`                              Eq2 := x + 3 z = 2`
 from where we obtain that,

> x := 2 - 3*z:
 finally the first equation is,

> Eq1 := w - 7*z = 8;

`                              Eq1 := w - 7 z = 8`
 from where we obtain that,

> w := 8 + 7*z:
 so there are infinitely many solutions to the system. All the points on the straight line

> {[w,x,y,z]};

`                        {[8 + 7 z, 2 - 3 z, -5 - z, z]}`
 where z is an arbitrary scalar. Ofcourse there are faster ways to get this answer with maple... here is one line solution,

> backsub(matrix(3,5,[1,0,0,-7,8, 0,1,0,3,2, 0,0,1,1,-5]));

`                 [8 + 7 _t[1], 2 - 3 _t[1], -5 - _t[1], _t[1]]`
 _t[1] is maple funny way (to make sure the name is not being used by you) of writing the free scalar parameter z

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>