SOLUTION: Use Cramer's rule to solve, please! -x-3y+z=7 3x +z=-1 3y+z=5

Algebra ->  Matrices-and-determiminant -> SOLUTION: Use Cramer's rule to solve, please! -x-3y+z=7 3x +z=-1 3y+z=5      Log On


   



Question 824928: Use Cramer's rule to solve, please!
-x-3y+z=7
3x +z=-1
3y+z=5

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
-x-3y+z=7
3x +z=-1
3y+z=5
Put in 1 or -1 coefficients, and +0y for the missing term in
the 2nd equation.  Put 0x for the missing term in the third
equation:

system%28-1x-3y%2B1z=red%287%29%2C3x%2B0y%2B1z=red%28-1%29%2C0x%2B3y%2B1z=red%285%29%29

Form the denominator determinant by using only the coefficients
to the left of the = sign.  You don't use the red numbers in D,
but you will in the others:

D=abs%28matrix%283%2C3%2C%0D%0A-1%2C-3%2C+1%2C%0D%0A3%2C+0%2C+1%2C%0D%0A0%2C+3%2C+1%29%29 = 21

[Do you know how to evaluate a 3x3 determinant?
If not, post again asking how to.  I will assume
you already know how.  That's where I got the 21.]

x is the FIRST unknown and so D%5Bx%5D is just like D
except that the FIRST column is replaced by the three red numbers,
which I made up: 

 = -42

y is the SECOND unknown and so D%5By%5D is just like D
except that the SECOND column is replaced by the three red numbers: 


 = 0

z is the THIRD unknown and so D%5Bz%5D is just like D
except that the THIRD column is replaced by the three red numbers: 

 = 105

So:
D=21
D%5Bx%5D=-42
D%5By%5D=0
D%5Bz%5D=105

Then the solutions are given by:

x=D%5Bx%5D%2FD=%28-42%29%2F21 = -2
y=D%5By%5D%2FD=%280%29%2F21 = %220%22
z%7D%7D=%7B%7B%7BD%5Bz%5D%2FD=%28105%29%2F21 = 5

Edwin