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To check that the solution is correct, enter the matrix A and multiply times this
solution vector (example in algebraic mode):
Under-determined system
The system of linear equations
2x
1
+ 3x
2
–5x
3
= -10,
x
1
– 3x
2
+ 8x
3
= 85,
can be written as the matrix equation A⋅x = b, if
This system has more unknowns than equations, therefore, it is not uniquely
determined. We can visualize the meaning of this statement by realizing that
each of the linear equations represents a plane in the three-dimensional
Cartesian coordinate system (x
1
, x
2
, x
3
). The solution to the system of
equations shown above will be the intersection of two planes in space. We
know, however, that the intersection of two (non-parallel) planes is a straight
line, and not a single point. Therefore, there is more than one point that satisfy
the system. In that sense, the system is not uniquely determined.
Let’s use the numerical solver to attempt a solution to this system of equations:
‚Ï ˜˜˜ @@OK@@ . Enter matrix A and vector b as illustrated in the
previous example, and press @SOLVE when the X: field is highlighted:
.
85
10
,,
831
532
3
2
1
⎥
⎦
⎤
⎢
⎣
⎡
−
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
−
−
= bxA and
x
x
x