Question 934976
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3x-y+2z=13 
2x+y-z=3 
x+3y-5z=-8

Below is one solved exactly like yours. Use it as a model to solve 
yours.

Whenever one of your equations has x with coefficient 1 or -1,
it makes it easier to write that equation first. Since your 
3rd equation has x with an understood coefficient of 1, it will 
be easier if you write it first, like this, swapping the 1st and 
3rd equations so that it is at the top.

x+3y-5z=-8
2x+y-z=3
3x-y+2z=13

BTW, yours has solution (x,y,z) = (3,-2,1)
  
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Here's one just like yours. Just use your numbers instead:

x+y+z=6
2x-y+z=3
x+2y-3z=-4

To solve using Cramer's rule
 
Write in all the 1 and -1 coefficients:

{{{system(1x+1y+1z=red(6),
2x-1y+1z=red(3),
1x+2y-3z=red(-4))}}}

There are 4 columns,
 
1. The column of x-coefficients {{{matrix(3,1,1,2,1)}}}
 
2. The column of y-coefficients {{{matrix(3,1,1,-1,2)}}}
 
3. The column of z-coefficients {{{matrix(3,1,1,1,-3)}}} 
 
4. The column of constants:     {{{red(matrix(3,1,6,3,-4))}}}
 
There are four determinants:
 
1. The determinant {{{D}}} consists of just the three columns
of x, y, and z coefficients. in that order, but does not
contain the column of constants.
 
{{{D=abs(matrix(3,3,1,1,1,2,-1,1,1,2,-3))}}}. 
 
It has value {{{D=13}}}.  I'm assuming you know how to find the
value of a 3x3 determinant, for that's a subject all by itself.
If you don't know how, post again asking how. 
 
2. The determinant {{{D[x]}}} is like the determinant {{{D}}}
except that the column of x-coefficients is replaced by the
column of constants.  {{{D[x]}}} does not contain the column 
of x-coefficients.
 
{{{D[x]=abs(matrix(3,3,red(6),1,1,red(3),-1,1,red(-4),2,-3))}}}.
 
It has value {{{D[x]=13}}}.
 
3. The determinant {{{D[y]}}} is like the determinant {{{D}}}
except that the column of y-coefficients is replaced by the
column of constants.  {{{D[y]}}} does not contain the column 
of y-coefficients.
 
{{{D[y]=abs(matrix(3,3,1,red(6),1,2,red(3),1,1,red(-4),-3))}}}.
 
It has value {{{D[y]=26}}}.
 
4. The determinant {{{D[z]}}} is like the determinant {{{D}}}
except that the column of z-coefficients is replaced by the
column of constants.  {{{D[z]}}} does not contain the column 
of z-coefficients.
 
{{{D[z]=abs(matrix(3,3,1,1,red(6),2,-1,red(3),1,2,red(-4)))}}}.
 
It has value {{{D[z]=39}}}.
 
Now the formulas for x, y and z are
 
{{{x=D[x]/D=13/13=1}}}
{{{y=D[y]/D=26/13=2}}}
{{{z=D[z]/D=39/13=3}}}

[To read about the 18th century Swiss mathematician Gabriel Cramer 
who invented Cramer's rule, go here: 
http://en.wikipedia.org/wiki/Gabriel_Cramer
 
Edwin</pre>