SOLUTION: I don't understand the steps of the Matrix for these equations. Could you please show me? 7x+8y-z=9 x-2y-5z=-31 -7x+y+z=0 Your help is highly appreciated! Thank you so muc

Put 1's in front of the single letters

 

Separate into terms, erasing the plus signs, keeping
the minus signs as negative signs:


 

Erase all the letters and replace the equal signs
with "|"'s:

 

Erase the brace and put parentheses around it:



Now we want to end up with a matrix like this,
with three zeros on the the bottom left, and
numbers everywhere else:



Start with this:



Swap the rows so that the smallest number in absolute
value in the first column is on the far left of the 
top row.  Since 1 is the smallest number in absolute
value in row 1, I will swap rows 1 and 2:



Now we will add -7 times the top row to the 2nd row,
to get a zero where the 7 is. It's easier if you
write -7 to the left of the top row and 1 to the left
of the second row,and write that equal to a new matrix
with the same 1st and 3rd rows, with a blank middle row:



Then you can easily fill in the blank row term by term as:



Since all the numbers in the middle row are even, we can
multiply it through by :



Now we will add 7 times the top row to the 3rd row,
to get a zero where the -7 is. It's easier if you
write 7 to the left of the top row and 1 to the left
of the bottom row,and write that equal to a new matrix
with the same 1st and 2nd rows, with a blank bottom row:



Then you can easily fill in the blank row term by term as:



---

Now we will add 13 times the middle row to 11 times
the 3rd row, to get a zero where the -13 is. It's 
easier if you write 13 to the left of the middle row 
and 11 to the left of the bottom row,and write that 
equal to a new matrix with the same 1st and 2nd rows,
with a blank bottom row:



Then you can easily fill in the blank row term by term as:



The bottom row can be multiplied through by 






Now we put the letters back as we took them out, and
put equal signs where the "|"'s are:



So we have this system:



Now we do what is called "back-substitution":

Substitute  into the middle equation:



Finally substitute both  and  in
the top equation:




So , , .

Edwin