Question 473485
What is w,x,y, and z when using matrices to solve the following system?
<pre>

The other tutor just gave the answer.

1w-3x-2y+1z=-3
2w-7x-1y+2z= 1
3w-7x-3y+3z=-5
5w+1x+4y-2z=18

Erase the letters, replace the ='s by |'s,
put parentheses around:

{{{(matrix(4,6,
1,-3,-2,1,"|",-3,
2,-7,-1,2,"|",1,
3,-7,-3,3,"|",-5,
5,1,4,-2,"|",18))}}}

The idea is to end up with a matrix that looks like this:


{{{(matrix(4,6,
1,X,X,X,"|",X,
0,1,X,X,"|",X,
0,0,1,X,"|",X,
0,0,0,1,"|",X))}}}

where there are various numbers where the X's are:

{{{(matrix(4,6,
1,-3,-2,1,"|",-3,
2,-7,-1,2,"|",1,
3,-7,-3,3,"|",-5,
5,1,4,-2,"|",18))}}}

Multiply the 1st row by -2, and add the second row

{{{matrix(4,6,
-2, 6, 4,-2,"|",6,
 2, -7,-1, 2,"|",1,
"--","--","--","--","","--", 
0,-1, 3, 0,"|",7)}}} 

Replace the 2nd row by the result:

{{{(matrix(4,6,
1,-3,-2,1,"|",-3,
0,-1,3,0,"|",7,
3,-7,-3,3,"|",-5,
5,1,4,-2,"|",18))}}}

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

Multiply the 1st row by -3, and add the 3rd row

{{{matrix(4,6,
-3, 9, 6,-3,"|",9,
3,-7,-3,3,"|",-5,
"--","--","--","--","","--", 
0,2, 3, 0,"|",4)}}} 

Replace the 3rd row by the result:

{{{(matrix(4,6,
1,-3,-2,1,"|",-3,
0,-1,3,0,"|",7,
0,2,3,0,"|",4,
5,1,4,-2,"|",18))}}}

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

Multiply the 1st row by -5, and add the 4th row

{{{matrix(4,6,
-5, 15, 10,-5,"|",15,
5,1,4,-2,"|",18,
"--","--","--","--","","--", 
0,16, 14,-7,"|",33)}}} 

Replace the 3rd row by the result:

{{{(matrix(4,6,
1,-3,-2,1,"|",-3,
0,-1,3,0,"|",7,
0,2,3,0,"|",4,
0,16, 14,-7,"|",33))}}}

Multiply the 2nd row by -1 to get a 1 in the 2nd row, 2nd column:

{{{(matrix(4,6,
1,-3,-2,1,"|",-3,
0,1,-3,0,"|",-7,
0,2,3,0,"|",4,
0,16, 14,-7,"|",33))}}}

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

Multiply the 2nd row by -2, and add the 3rd row

{{{matrix(4,6,
0,-2, 6,0,"|",14,
0,2,3,0,"|",4,
"--","--","--","--","","--", 
0,0, 9,0,"|",18)}}} 

Replace the 3rd row by the result:

{{{(matrix(4,6,
1,-3,-2,1,"|",-3,
0,1,-3,0,"|",-7,
0,0,9,0,"|",18,
0,16, 14,-7,"|",33))}}}

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

Multiply the 2nd row by -16, and add the 4th row

{{{matrix(4,6,
0,-16,48,0,"|",112,
0,16,14,-7,"|",33,
"--","--","--","--","","--", 
0,0,62,-7,"|",145)}}} 

Replace the 4th row by the result:

{{{(matrix(4,6,
1,-3,-2,1,"|",-3,
0,1,-3,0,"|",-7,
0,0,9,0,"|",18,
0,0,62,-7,"|",145))}}}

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

Multiply the 3rd row by 1/9 to get a 1 in the 3rd row, 3rd column:



{{{(matrix(4,6,
1,-3,-2,1,"|",-3,
0,1,-3,0,"|",-7,
0,0,1,0,"|",2,
0,0,62,-7,"|",145))}}}

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

Multiply the 3rd row by -62, and add the 4th row

{{{matrix(4,6,
0,0,-62,0,"|",-124,
0,0,62,-7,"|",145,
"--","--","--","--","","--", 
0,0,0,-7,"|",21)}}} 

Replace the 4th row by the result:

{{{(matrix(4,6,
1,-3,-2,1,"|",-3,
0,1,-3,0,"|",-7,
0,0,1,0,"|",2,
0,0,0,-7,"|",21))}}}

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

Multiply the 4th row by -1/7 to get a 1 in the 4th row, 4th column:

{{{(matrix(4,6,
1,-3,-2,1,"|",-3,
0,1,-3,0,"|",-7,
0,0,1,0,"|",2,
0,0,0,1,"|",-3))}}}

Now we have the matrix in the form, called "triangular form",
we convert it back to a system of equations:

{{{system(
1w-3x-2y+1z=-3,
0w+1x-3y+0z=-7,
0w+0x+1y+0z=2,
0w+0x+0y+1z=-3)}}}

Get rid of the unnecessary or understood 0 terms and 1's

{{{system(
w-3x-2y+z=-3,
x-3y=-7,
y=2,
z=-3)}}}

We have the values for y and z, so we substitute y=2
into the 2nd equation:

{{{matrix(4,1,
x-3y=-7,
x-3(2)=-7,
x-6=-7,
x=-1)}}}

Finally we substitute x=-1, y=2, z=-3 into the 1st equation:

{{{matrix(5,1,
w-3x-2y+z=-3,
w-3(-1)-2(2)+(-3)=-3,
w+3-4-3=-3,
w-4=-3,
w=1)}}}

(w,x,y,z) = (1,-1,2,-3)

Edwin</pre>