Question 626268
Using the matrices find a general solution
 <pre>
{{{system(2x-1y+3z=5,
 -1x+4y+4z=-1)}}}

Convert to this matrix:

{{{(matrix(2,5,
2,-1,3,"|",5,
-1,4,4,"|",-1))}}}

We want to use row operations to get it looking like this: {{{(matrix(2,5,
1,0,"#","|","#",
0,1,"#","|","#"))}}}

First we get the two zeros:

To get a 0 where the -1 is on the bottom left, multiply
the second row by 2

{{{(matrix(2,5,
2,-1,3,"|",5,
-2,8,8,"|",-2))}}}

and add the first row to the 2nd row

{{{(matrix(2,5,
2,-1,3,"|",5,
0,7,11,"|",3))}}}

To get a 0 where the remaining -1 is on the first row, multiply
the first row by 7

{{{(matrix(2,5,
14,-7,21,"|",35,
0,7,11,"|",3))}}}

and add the second row to the first row:

{{{(matrix(2,5,
14,0,32,"|",38,
0,7,11,"|",3))}}}

Now we get the two 1's

Divide the first row through by 14, and the second row
through by 7

{{{(matrix(2,5,
14/14,0/14,32/14,"|",38/14,
0/7,7/7,11/7,"|",3/7))}}}

and reduce:

{{{(matrix(2,5,
1,0,16/7,"|",19/7,
0,1,11/7,"|",3/7))}}}

convert back to a system of equations:

{{{system(
1x+0y+expr(16/7)z=19/7,
0x+1y+expr(11/7)z=3/7)}}}

or

{{{system(
x=19/7-expr(16/7)z,
y=3/7-expr(11/7)z)}}}

Solution: choose any value k for z then annex z=k to the system:

{{{system(
x=19/7-expr(16/7)k,
y=3/7-expr(11/7)k,z=k)}}}

or

(x,y,z) = ({{{19/7-expr(16/7)k}}},{{{3/7-expr(11/7)k}}},k)

Edwin</pre>