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 |