SOLUTION: Suppose you want to use Cramer's rule to find the value of z in the following system.. write the determinants you would need to evaluate. {{{system(-7x+3y+9z=12, 5x +3z=8, 4x

Algebra ->  Algebra  -> Matrices-and-determiminant -> SOLUTION: Suppose you want to use Cramer's rule to find the value of z in the following system.. write the determinants you would need to evaluate. {{{system(-7x+3y+9z=12, 5x +3z=8, 4x      Log On

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Question 168729: Suppose you want to use Cramer's rule to find the value of z in the following system.. write the determinants you would need to evaluate.
system%28-7x%2B3y%2B9z=12%2C%0D%0A5x+%2B3z=8%2C%0D%0A4x-6y%2Bz=-2%29
Answer by Edwin McCravy(6939) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose you want to use Cramer's rule to find the value of z in the following system.. write the determinants you would need to evaluate.
system%28-7x%2B3y%2B9z=12%2C%0D%0A5x+%2B3z=8%2C%0D%0A4x-6y%2Bz=-2%29


You just asked for z, but I'll give you the determinant for all 
three unknowns:

x=D%5Bx%5D%2FD=abs%28matrix%283%2C3%2C12%2C3%2C9%2C8%2C0%2C3%2C-2%2C-6%2C1%29%29%2Fabs%28matrix%283%2C3%2C-7%2C3%2C9%2C5%2C0%2C3%2C4%2C-6%2C1%29%29=%28-258%29%2F%28-375%29=86%2F375


y=D%5By%5D%2FD=abs%28matrix%283%2C3%2C-7%2C12%2C9%2C5%2C8%2C3%2C4%2C-2%2C1%29%29%2Fabs%28matrix%283%2C3%2C-7%2C3%2C9%2C5%2C0%2C3%2C4%2C-6%2C1%29%29=%28-392%29%2F%28-375%29=392%2F375

z=D%5Bz%5D%2FD=abs%28matrix%283%2C3%2C-7%2C3%2C12%2C5%2C0%2C8%2C4%2C-6%2C-2%29%29%2Fabs%28matrix%283%2C3%2C-7%2C3%2C9%2C5%2C0%2C3%2C4%2C-6%2C1%29%29=%28-570%29%2F%28-375%29=38%2F25

The only answer you want is this last one:

z=D%5Bz%5D%2FD=%28abs%28matrix%283%2C3%2C-7%2C3%2C12%2C5%2C0%2C8%2C4%2C-6%2C-2%29%29%29%2F%28abs%28matrix%283%2C3%2C-7%2C3%2C9%2C5%2C0%2C3%2C4%2C-6%2C1%29%29%29

Notice that the denominators are all the same and they are made
from the coefficients of the unknowns.  Notice that each numerator
determinant is just like the denominator except for 

the 1st column of the 1st unknown x,
the 2nd column of the 2nd unknown y, and 
the 3rd column of the 3rd unknown z.  

What replaces these is the column of constants matrix%283%2C1%2C12%2C8%2C-2%29 on the right of 
the equal signs, which does not appear in the denominator.

Edwin