Question 315481
{{{system(4x - 3y +z +w = 21,
-2x - y + 2z + 7w = 2,
10x - 5z - 20w = 15)}}}
<pre><b>
Write the augmented matrix:
{{{(matrix(3,6,
 4, -3,   1,   1, "|", 21,
-2, -1,   2,   7, "|", 2,
10,  0,   -5, -20, "|", 15))}}}

1*R1+2*R2->R2

{{{(matrix(3,6,
 4, -3,   1,    1, "|", 21,
 0, -5,   5,   15, "|", 25,
10,  0,   -5, -20, "|", 15))}}}

-5*R1+2*R3->R3

{{{(matrix(3,6,
 4, -3,   1,    1, "|",  21,
 0, -5,   5,   15, "|",  25,
 0, 15, -15,  -45, "|", -75))}}}

{{{-1/5}}}*R2->R2
{{{1/15}}}*R3->R3

{{{(matrix(3,6,
 4, -3,   1,    1, "|",  21,
 0,  1,  -1,   -3, "|",  -5,
 0,  1,  -1,   -3, "|",  -5))}}}

 3*R2+1*R1->R1
-1*R2+1*R3->R3

{{{(matrix(3,6,
 4,  0,  -2,   -8, "|",   6,
 0,  1,  -1,   -3, "|",  -5,
 0,  0,   0,    0, "|",   0))}}}

{{{1/4}}}*R1->R1

{{{(matrix(3,6,
 1,  0, -1/2,  -2, "|",  3/2,
 0,  1,  -1,   -3, "|",  -5,
 0,  0,   0,    0, "|",   0))}}}

Write as a system of equations:

{{{system(1x + 0y -(1/2)z -2w = 3/2,
0x + 1y - 1z - 3w = -5,
0x + 0y+0z +0w = 0)}}}

Simplify

{{{system(x -(1/2)z -2w = 3/2,
  y - z - 3w = -5)}}}

Solve only for the first two variables, since there are
only two equations:

{{{system(x = 3/2 +(1/2)z+2w,
  y = -5+z+3w)}}}

Let z be arbitrary variable {{{a}}} and let w be arbitrary variable {{{b}}}

{{{system(x = 3/2 +(1/2)a+2b,
  y = -5+a+3b,z=a,w=b)}}} 

or

(x, y, z, w) = ({{{3/2+(1/2)a+2b}}},-5+a+3b,a,b)

Edwin</pre>