Question 908650
coefficient matrix
.25,.5,1
4,2,1
1,1,1
coefficient matrix
{{{
(
 matrix( 3,3, 
.25,.5,1,
4,2,1,
1,1,1
 )
)

}}}
augmented decimals
.25,.5,1|5.75
4,2,1|2
1,1,1|7

{{{
(
 matrix( 3, 4, 
.25,.5,1,5.75,
4,2,1,2,
1,1,1,7
 )
)
}}}
augmented fractions
{{{
(
 matrix( 3, 4, 
1/4,1/2,1,5&3/4,
4,2,1,2,
1,1,1,7
 )
)
}}}

.25,.5,1,5.75
4,2,1,2
1,1,1,7

divide row 1 by 0.25
1,2,4,23
4,2,1,2
1,1,1,7

add  down (-4) *row 1 to row 2
1,2,4,23
0,-6,-15,-90
1,1,1,7

add  down (-1) *row 1 to row 3
1,2,4,23
0,-6,-15,-90
0,-1,-3,-16

divide row 2 by -6
1,2,4,23
0,1,-15/-6,15
0,-1,-3,-16

add  down (1) *row 2 to row 3
1,2,4,23
0,1,5/2,15
0,0,-1/2,-1

divide row 3 by -1/2
1,2,4,23
0,1,5/2,15
0,0,1,2

We now have the value for the last variable.
We will work our way up and get the other solutions.

add up  (-5/2) *row 3 to row 2
1,2,4,23
0,1,0,10
0,0,1,2

add up  (-4) *row 3 to row 1
1,2,0,15
0,1,0,10
0,0,1,2

add up  (-2) *row 2 to row 1
1,0,0,-5
0,1,0,10
0,0,1,2

final
1,0,0,-5
0,1,0,10
0,0,1,2
"-5","10","2"
(-5,10,2)