solve using cramers rule:
2x-3y-3z=25
3x+1y-1z=-5
5x-2y-4z=32
I am very confused with this problem...please help!
Cramer's rule can only be used to solve independent
systems. If the system is dependent or inconsistent,
then Cramer's rule cannot be used to solve the system.
There are 4 determinants that must be made with Cramer's
rule: D, Dx, Dy, and Dz.
If it turns out that D is not 0, then
x = Dx/D, y = Dy/D, and z = Dz/D.
However if it turns out that D = 0, then Cramer's rule
cannot be used, because either the system is dependent
and has INFINITELY MANY solutions or else it is
inconsistent and has NO solutions.
First we make D. Here is how:
Start with the system:
2x-3y-3z=25
3x+1y-1z=-5
5x-2y-4z=32
Erase all the letters, the equal signs, and the
numbers on the right of the equal sign
2 -3 -3
3 +1 -1
5 -2 -4
Put bars around it:
|2 -3 -3|
|3 +1 -1|
|5 -2 -4|
That's the determinant called D.
Do you know how to evaluate a 3x3 determinant? If
you don't post again and ask how to evaluate it. I
will assume you know how.
It turns out that D = 0, so Cramer's rule cannot be
used to solve this system. The system is either
dependent and has INFINITELT MANY solutions, or it
is inconsistent and has NO solutions. To find out
which it is, we form Dx, Dy and Dz. If they are ALL
0, then the system is dependent and has infinitely
many solutions. If ANY ONE of Dx, Dy, or Dz is NOT 0,
then the system is inconsistent and there are NO solutions.
x is the FIRST unknown that appears, so
Dx is just like D except that the FIRST
column is replaced by the column of
constants. These are the numbers on the
right of the equal signs in the system,
i.e., the red numbers below:
2x-3y-3z=25
3x+1y-1z=-5
5x-2y-4z=32
Here is Dx:
|25 -3 -3|
|-5 +1 -1|
|32 -2 -4|
y is the SECOND unknown that appears, so
Dy is just like D except that the SECOND
column is replaced by the column of
constants.
Here is Dy:
|2 25 -3|
|3 -5 -1|
|5 32 -4|
z is the THIRD unknown that appears, so
Dz is just like D except that the THIRD
column is replaced by the column of
constants.
Here is Dz:
|2 -3 25|
|3 +1 -5|
|5 -2 32|
Evaluate those
Dx = 72, Dy = -84, Dz = 132
They certainly are NOT all 0. In fact,
none of them are, so the system is
INCONSISTENT and there is no solution.
Edwin
