SOLUTION: solve the following system of equtions and give the general solution in parametric vector form. x+y-2z=6 3y+z-w=3 -2x+y+2z-w=9

Algebra ->  College  -> Linear Algebra -> SOLUTION: solve the following system of equtions and give the general solution in parametric vector form. x+y-2z=6 3y+z-w=3 -2x+y+2z-w=9      Log On


   

Question 86464This question is from textbook
: solve the following system of equtions and give the general solution in parametric vector form.
x+y-2z=6
3y+z-w=3
-2x+y+2z-w=9 This question is from textbook

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

solve the following system of equtions and give the general
solution in parametric vector form. 

  x + y - 2z     = 6
     3y +  z - w = 3
-2x + y + 2z - w = 9

Associated with this linear system is the augmented matrix

%28matrix%283%2C5%2C1%2C1%2C-2%2C0%2C6%2C0%2C3%2C1%2C-1%2C3%2C-2%2C1%2C2%2C-1%2C9%29%29

Reducing it to row reduced echelon form:



So the general solution is 

%28matrix%284%2C1%2Cx%2Cy%2Cz%2Cw%29%29 = %28matrix%284%2C1%2C-9-%281%2F3%29w%2C3%2B%281%2F3%29w%2C-6%2Cw%29%29

Write the -6 as -6+0w, and the w as 0 + 1w

%28matrix%284%2C1%2Cx%2Cy%2Cz%2Cw%29%29 = %28matrix%284%2C1%2C-9-%281%2F3%29w%2C3%2B%281%2F3%29w%2C-6%2B0w%2C0%2B1w%29%29

Write the matrix on the right as the sum of two matrices:

%28matrix%284%2C1%2Cx%2Cy%2Cz%2Cw%29%29 = %28matrix%284%2C1%2C-9%2C3%2C-6%2C0%29%29 + %28matrix%284%2C1%2C-%281%2F3%29w%2C%281%2F3%29w%2C0w%2C1w%29%29

Now factor out a scalar w from the right-hand matrix:

%28matrix%284%2C1%2Cx%2Cy%2Cz%2Cw%29%29 = %28matrix%284%2C1%2C-9%2C3%2C-6%2C0%29%29 + w%28matrix%284%2C1%2C-1%2F3%2C1%2F3%2C0%2C1%29%29

That's fine in that form, however, it looks neater when you factor out
1/3 from the matrix on the right:

%28matrix%284%2C1%2Cx%2Cy%2Cz%2Cw%29%29 = %28matrix%284%2C1%2C-9%2C3%2C-6%2C0%29%29 + %281%2F3%29w%28matrix%284%2C1%2C-1%2C1%2C0%2C3%29%29

Edwin