Question 643660
<pre>
This is an under-determined system because there are fewer equations
than there are variables. 

2x -  y + 3z =  5
-x + 4y + 4z = -1

Put in 1's

 2x - 1y + 3z =  5
-1x + 4y + 4z = -1

Write the system as a matrix [by eliminating the letters, putting | for
equal signs and enclosing in brackrts]:

[ 2  -1    3 |  5]
[-1   4    4 | -1]

We want to use row operations
to make it look like this:

[ 1  0  # |  #]
[ 0  1  # |  #]

where there are numbers where the #'s are
but 1's and 0's as indicated:

[ 2  -1    3 |  5]
[-1   4    4 | -1]

Multiply the 2nd row by 2

[ 2  -1   3  |  5]
[-2   8   8  | -2]

Add the 1st row to the 2nd row to
get a 0 in the lower left corner:

[ 2  -1   3  | 5]
[ 0   7  11  | 3]

Multiply the 1st row by 7

[14  -7  21  |35]
[ 0   7  11  | 3]

Add the 2nd row to the 1st row to
get a 0 where the -7 is:

[14   0  32 |38]
[ 0   7  11 | 3]

Get a 1 where the 14 is by dividing the 
1st row through by 14

Get a 1 where the 7 is by dividing the
2nd row through by 7

[ 1   0 {{{32/14}}} | {{{38/14}}}]
[ 0   1 {{{11/7}}} | {{{3/7}}}]

Reduce the fractions:

[ 1   0 {{{16/7}}} | {{{19/7}}}]
[ 0   1 {{{11/7}}} | {{{3/7}}}]

Write the matrix as a system of equations by putting
the letters and the equal signs back in:

 1x + 0y + {{{16/7}}}z = {{{19/7}}}
 0x + 1y + {{{11/7}}}z =  {{{3/7}}}

Simplifying:

 x + {{{16/7}}}z = {{{19/7}}}
 y + {{{11/7}}}z =  {{{3/7}}}

Solving for x and y

 x = {{{19/7}}} - {{{16/7}}}z
 y =  {{{3/7}}} - {{{11/7}}}z

Now you can choose any arbitrary number for z to equal,
say z = a

 x = {{{19/7}}} - {{{16/7}}}z
 y =  {{{3/7}}} - {{{11/7}}}z
 z = a

And write the general solution this way, letting z = a:

 x = {{{19/7}}} - {{{16/7}}}a
 y =  {{{3/7}}} - {{{11/7}}}a
 z = a

Or as an ordered triple:

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

That's the general solution:

--------------------------------------------------------------

Then if you want some sample solutions, you can pick various
values for a. If you pick a= 0, you have this solution:

(x, y, z) = ({{{19/7}}}, {{{3/7}}}, 0)

which has fractions.

But by picking the right values for a=z you can get integer solutions.
Here are some particular solutions:

(x,y,z) = (5,2,-1)
(x,y,z) = (21,13,-8)
(x,y,z) = (37,24,-15)
(x,y,z) = (53,35,-22)
(x,y,z) = (69,46,-29)
(x,y,z) = (85,57,-36)
(x,y,z) = (101,68,-43)
(x,y,z) = (117,79,-50)

Edwin</pre>